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THOMSON'S     NEW     GRADtD     StRlES. 


NEW 


PRACTICAL 


AEITHMETIC: 


FOR 


GRAMMAR    DEPARTMENTS 


By  JAMRS  B.  THOMSOS,  LL.  D., 

^UTHOB  OF  DAT  &  THOMSON  S  ARITHMETICAL  SERIES  ;  EDITOR  OF  DaT'S  SCUOO'4 
ALGEBRA,  LSGENDRS'S  GEOHETRT,  ETC. 


TfllETY-FIFTH    EDITION. 


NEW    YORK: 

CLARK  &  MAYNARD,  PUBLISHERS. 

5  barclay  street. 

Chicago:    46  Madisoj^  Street. 


-••■': .•ipifois.ok^s vMathematical  Series 

I.  A  Graded  Series  of  Arithmetics,  in  three  Books,  viz.  : 

New  Illustrated  Table  Book,   or   Juvenile  Arithmetic.      With  oral 
and  slate  exercises.     (For  beginners.)     128  pp. 

New    Rudiments    of   Arithmetic.      Combining  Mental   with   Written 
Arithmetic.     (For  Intermediate  Classes.)    224  pp. 

New  Practical  Arithmetic.    Adapted  to  a  complete  business  education. 
(For  Grammar  Departments.)    384  pp. 

II.  Independent  Books. 

Key  to  New  Practical  Arithmetic.      Containing  many  valuable  sug- 
gestions.    (For  teachers  only.)     168  pp. 

New   Mental   Arithmetic.      Containing    the    Simple   and    Compound 
Tables.     (For  Primary  Schools.)     144  pp. 

Complete   Intellectual   Arithmetic.      Specially  adapted  to  Classes  in 
Grammar  Schools  and  Academies.     168  pp. 

III.  Supplementary  Course, 

New  Practical  Algebra.     Adapted  to  High  Schools  and  Academies. 
312  pp. 

Key  to  New  Practical  Algebra.     With  full  solutions.     (For  teachers 
only.)    224  pp. 

New  Collegiate  Algebra. 

Complete  Higher  Arithmetic.     (In  preparation.) 

\*  Each  book  of  tJie  Series  is  complete  in  itself. 


Copyright,  1872,  by  James  B.  Thomson. 


EDUCATION  DEPT 


PREFACE 


THE  New  Practical  Arithmetic  now  offered  to  the 
public,  is  the  third  and  last  of  the  works  which 
constitute  the  author's  "  New  Graded  Series." 

The  old  "Practical"  was  issued  in  1845,  and  revised  in 
1853.  Since  that  time  important  changes  have  taken 
lilace  in  the  commercial  world.  These  changes  necessarily 
affect  business  calculations,  and  demand  corresponding 
modifications  in  text  books. 

To  meet  this  demand,  the  "New  Graded  Series"  was 
undertaken.  Each  part  of  the  series  has  been  reinvesti- 
gated and  rewritten; — the  whole  being  readjusted  upon 
the  graded  plan,  and  brought  down  to  the  present  wants. 

Among  the  objects  aimed  at  in  the  present  work  are 
the  following : 

I.  To  make  the  definitions  clear,  concise,  and  com,' 


2.  To  present  the  principles  of  the  science  in  a  series 
of  distinct  and  consecutive  propositions. 

3.  To  lead  the  mind  of  the  pupil,  through  the  analysis 
of  the  examples  immediately  following  the  respective 
propositions,  to  discover  the  principles  by  w"hich  all  similar 
examples  are  solved,  and  enable  him  to  sum  up  the  prin- 
ciples thus  developed, into  a  brief,  comprehensive  rule. 

4.  The  "how"  and  the  "why"  are  fully  explained. 

5.  Great  pains  have  been  taken  to  ascertain  the  Standard 
Weights  and  Measures  authorized  by  the  Government;  and 
to  discard  from  the  Tables  such  denominations  as  are 
ohsolete,  or  not  used  in  this  country.* 

*  Laws  of  Congress ;  Profs,  Haspler,  Bache  and  Egleston ;  Reports  of  Supts. 
Harrison  and  Calkins ;  also  of  the  Committee  on  Weights  and  Measures  of  the 
Paris  Exposition,  1867. 


ivi34:y349 


PREFA  CE. 

6.  The  Metric  System  is  accompanied  with  brief  and 
appropriate  explanations  for  reducing  it  to  practice.  Its 
simplicity  and  comprehensiveness  have  secured  its  use  in 
the  natural  sciences  and  commerce  to  such  an  extent  in 
this  and  foreign  countries,  that  no  student  can  be  said  to 
have  a  finished  education,  without  a  knowledge  of  it. 

7.  Particular  attention  has  been  paid  to  the  develop- 
ment of  Analysis,  the  grand  common  sense  rule  which 
business  men  intuitively  adopt  as  they  enter  upon 
practical  life. 

8.  The  examples  are  new  and  abundant ; — being  drawn 
from  the  various  industrial  arts,  commerce,  science,  etc. 

9.  The  arrangement  of  the  matter  upon  the  page,  and  the 
typogi-aphy,  have  also  received  due  attention.  Teachers 
who  deal  much  with  figures,  will  be  pleased  with  the 
adoption  of  the  Eranklin  type.  The  ease  with  which 
these  figures  are  read,  is  sufficiently  attested  by  their  use 
in  all  recent  Mathematical  Tables. 

Finally,  it  has  been  the  cardinal  object  to  adapt  the 
science  of  numbers  to  the  present  wants  of  the  farm,  the 
household,  the  workshop,  and  the  counting-room ; — in  a 
word,  to  incorporate  as  much  information  pertaining  to 
business  forms,  and  matters  of  science,  as  the  limits  of  the 
book  would  perm-it.  In  this  respect  it  is  believed  the 
work  is  unrivaled.  While  it  puts  forth  no  claim  to 
mathematical  paradoxes,  it  is  believed  teachers  will  find 
that  something  worthy  of  their  attention  is  gained,  in 
nearly  every  Article. 

In  conclusion,  the  author  tenders  his  most  cordial 
thanks  to  teachers  and  the  public  for  the  very  liberal 
patronage  bestowed  upon  his  former  Arithmetics,  known 
as  "  Day  and  Thomson's  Series."  It  is  hoped  the  "  New 
Graded  Series"  will  be  found  worthy  of  continued  favor. 

James  B.  Thomson. 
I5[ew  York,  Jvly.,  187^ 


CONTENTS. 


PAfiB 

N"umber, 9 

Rotation, lo 

Arabic  Notation, lo 

Koman  Notation, 15 

To  Express  Numbers  by  Letters,   -        -        -        -17 

JSTumerationy 17 

French  Numeration, 18 

English  Numeration, 20 

Addition^ 21 

When  the  Sum  of  each  Column  is  less  than  10,  -  23 

When  the  Sum  of  a  Column  is  10  or  more,    -        -  24 

Carrying  Illustrated, 24 

Drill  Columns, 29 

Subtraction, 31 

When  each  Figure  in  the  Subtrahend  is  less  than 

that  above  it, 32 

When   a  Figure  in  the   Subtrahend  is  greater 

than  that  above  it, 34 

Borrowing  Illustrated, 34 

Questions  for  Review,         _        .        _        -        -  38 

Multiplication, 40 

When  the  Multiplier  has  but  one  Figure,   -        -  43 

When  the  Multiplier  has  more  than  one  Figure,    -  45 

To  find  the  Excess  of  9s, 47 

Contractions,         -------49 


VI  COKTEN^TS. 

PASB 

Division^ 53 

The  Two  Problems  of  Division,  -  -  -  -  55 
Short  Division,  -  -  -.  -  -  -  '57 
Long  Division,  -        -        -        -        -        --        61 

Contractions, 65 

Questions  for  Review,  -  -  -  -  -  69 
General  Principles  of  Division,  -  -  -  -71 
Problems  and  Formulas  in  the  Fundamental  Eules,       7  2 

Analysis,         - 77 

Classification  and  Properties  of  Numbers,       -        -  80 

The  Complement  of  Numbers,   -        -        -        -  82 

Divisibility  of  Numbers, S^ 

Factor  Ing,       -       -       -       -        -        -       -  85 

Prime  Factors, 86 

Cancellation^ 88 

Greatest  Common  Divisor,     -        -        -        -        -  91 

Least  Common  Multiple, 96 

Fractions, 99 

To  find  a  Fractional  Part  of  a  Number,  -  -  102 
General  Principles  of  Fractions,  -  -  -  -  103 
Reduction  of  Fractions,      -        -        -        -        -       104 

A  Common  Denominator, iii 

The  Least  Common  Denominator,      -        -        -      112 

Addition  of  Fractions, 114 

Subtraction  of  Fractions, 117 

Multiplication  of  Fractions,  -  -  -  -  -  120 
General  Rule  for  multiplying  Fractions,     -        -       125 

Division  of  Fractions, 126 

General  Rule  for  dividing  Fractions,  -        -       131 

Questions  for  Review, 132 

Fractional  Relations  of  Numbers,       -        -        -       134 

J)ecinial  Fractions, 139 

Eeduction  of  Decimals, 144 

Addition  of  Decimals,   - 146 

Subtraction  of  Decimals,  -  -  -  -  -  148 
Multiplication  of  Decimals,  -  -  -  -  -  149 
Piyision  of  Decimals,  -        -        -        •        -       152 


C  0  K  T  E  K  T  S .  Vil 

PAGB 

United  States  Money, i54 

Addition  of  U.  S.  Money, 15S 

Subtraction  of  U.  S.  Money,  -  -  -  "'  -  15  9 
Multiplication  of  U.  S.  Money,  -  -  -  -  160 
Division  of  IT.  S.  Money,        -        -        -         -        -161 

Counting-room  Exercises, 163 

Making  out  Bills, ^164 

Business  Methods, ^^^ 

Cofnpound  Nmnhers, 171 

Money, ^7^ 

Weights, 175 

Measures  of  Extension, i77 

Measures  of  Capacity, 182 

Circular  Measure, -       184 

Measurement  of  Time,  -        -        -        -        -        -  186 

Reduction, 1^9 

Application  of  Weights  and  Measures,  -        -        -     i94 
Artificers' Work,       _-----        196 

Measurement  of  Lumber, ^9^ 

Denominate  Fractions, ^^^ 

Metric  Weights  and  Measures,  -       -        -  207 
Apphcation  of  Metric  Weights  and  Measures,     -       214 

Compound  Addition, -  216 

Compound  Subtraction, 219 

Compound  Multiplication, 224 

Compound  Division, 226 

Comparison  of  Time  and  Longitude,      -        -        -  227 

Percentage^ ^2° 

Notation  of  Per  Cent, 230 

Five  Problems  of  Percentage,     -        -        -        -      233 

Applications  of  Percentage,       -       -       -  241 

Commission  and  Brokerage,       ^        '        '        '      ^^l 
Account  of  Sales,  -----•"4 

Profit  and  Loss, ^"^l 

Interest,         ------  ^J 

Preliminary  Principles,     -        -        -        "        ■  55 

Six  Per  Cent  Method,  ------     25 


mi  CONTENTS. 

PAoa 
Method  by  Aliquot  Parts,      -        -        -        -        -259 

Method  by  Days, 260 

Partial  Payments,  -        -        -        -        -        -267 

Compound  Interest,  -        -        -        -        -        -273 

Discount, -        -276 

Banks  and  Bank  Discount,        -        -        -        -      278 
Stock  Investments,      -        -        -        -        -        -280 

Government  Bonds, 281 

Exchange,  .-.----    285 

Insurance,  -------      292 

Taxes,  ---------  295 

Duties,      --------      298 

Internal  Kevenue,  ------  300 

Equation  of  Payments,       -        -        -        -        -      301 

Averaging  Accounts,     -.----  304 

Ratio,        ---..---      307 
Proportion^         ---.---  309 

Simple  Proportion,     -        -        -        -        -        -311 

Simple  Proportion  by  Analysis,     -        -        -        -  Z'^Z 

Compound  Proportion,       -        -        -        -        -316 

Partitive  Proportion,      -        -        -        -        -        -319 

Partnership,        -         -     *   -        -        -        -         -       320 

Bankruptcy, 323 

Alligation,         -        -        -        -        -        -        -324 

TnroliitioTif 330 

Formation  of  Squares,        -        -        -        -        -      332 

UroJution,  --        -        -        -        -        -        -333 

Extraction  of  the  Square  Eoot,  -        -        -      335 

Applications  of  Square  Root,  -        -        -        -  33^ 

Formation  of  Cubes, 343 

Extraction  of  the  Cube  Root,         -        -        -        -  345 

Applications  of  Cube  Eoot,        -        -        -        -      349 

Arithmetical  JProgression,    -       -       -       -  350 

Geometrical  I*rogression,         -        -        -      ^^^ 

Mensuration, 355 

Miscellaneous  Examples,    -----      360 


ARITHMETIC 


Art.  1.   Arithtnetic  is  the  science  of  numbers. 

Aritlimetic  is  sometimes  said  to  be  both  a  science  and  an  art :  a 
science  when  it  treats  of  the  theory  and  properties  of  numbers ;  an 
art  when  it  treats  of  their  applications. 

Notes. — i.  The  term  arithmetic,  is  from  the  Greek  arithmetike, 
the  art  of  reckoning. 

2.  The  term  science,  from  the  Latin  scientia,  literally  signifies 
knowledge.  In  a  more  restricted  sense,  it  denotes  an  oi'derly  ar- 
rangement of  the  facts  and  principles  of  a  particular  branch  of 
knowledge. 

2.  Number  is  a  unit,or  a  collection  of  nnits. 

A  Unit  is  any  single  thing,  called  07ie.  One  and 
one  more  are  called  tivo ;  two  and  one  more  are  called 
three ;  three  and  one  more,  four,  etc.  The  terms  one, 
two,  three,  four,  are  properly  the  names  of  numbers,  but 
are  often  used  for  numbers  themselves. 

Note. — The  term  unit  is  fr9m  the  Latin  unus,  signifying  one. 

3.  The  TJnM  One  is  the  standard  by  which  all  num- 
bers are  measured.  It  may  also  be  considered  the  hase 
or  element  of  number.  For,  all  icliole  numbers  greater 
than  one  are  composed  of  ones.  Thus,  tivo  is  composed  of 
one  and  one.  Three  is  one  more  than  two ;  but  two,  we 
have  seen,  is  composed  of  ones ;  hence,  three  is,  and  so  on. 


Questions. — i.  What  is  arithmetic?  .  What  else  is  it  sometimes  said  to  be? 
When  a  science  ?  An  art  ?  2.  Number  ?  A  unit  ?  3.  The  tcandard  by  which 
numbers  are  measured  ?    The  base  or  element  of  number  * 


1.0  .  KOTATIOK. 

4.   Numbers  *are  either  abstract  or  concrete. 

An  Abstract  dumber  is  one  that  is  not  applied  to 
any  object;  as,  three,  five,  ten. 

A  Concrete  Number  is  one  that  is  applied  to  some 
object;  as,  five  peaches,  ten  books. 


NOTATION. 

5.  JS'otatioii  is  the  art  of  expressing  numbers  by 
figures,  letters,  or  other  numeral  characters. 

The  two  principal  methods  in  use  are  the  Arabic  and 
the  Roman. 

Note. — ^Numbers  are  also  expressed  bj  words  or  common  lan- 
guage;  but  this,  strictly  speaking,  is  not  Notation. 

ARABIC    NOTATION. 

6.  The  Arabic  Notation  is  the  method  of  express- 
ing numbers  by  certain  characters  called  figures. 

It  is  so  called,  because  it  was  introduced  into  Europe 
from  Arabia. 

7.  The  Arabic  figures  are  the  following  ten,  viz : 

1,     2,     3,    4,     5,     6,     7,     8,     9,     o. 

oiic,     two,    three,  four,    five,       six,  seven,  eight,    nine,  naught. 

The  first  nine  are  called  significant  figures,  or  digits ; 
the  last  one,  naught,  zero,  or  cipher. 

Notes. — i.  Tbe  first  nine  are  called  aignijicant  figures,  because 
each  always  expresses  a  number. 

2.  The  term  digit  is  from  the  Latin  digitus,  a  finger,  and  was 
applied  to  these  characters  because  they  were  employed  as  a  substi- 

4.  What  is  an  abstract  number?  Concrete?  5.  What  is  notation?  The 
principal  methods  in  use  ?  6.  Arabic  notation  ?  Why  so  called  ?  7.  How  many 
figures  does  it  employ  ?  What  are  the  £rst  nine  called  ?  The  last  one  ?  Note. 
Why  called  significant  figures  ?  Why  digits  ?  Meaning  of  digitus  ?  Why  is  the 
ki^t  called  naught  ?    Meaning  of  zero  ?    Of  cipher  ? 


NOTATION".  11 

tute  for  the  ^w^ers  upon  wliicli  the  ancients  used  to  reckon.  The 
term  originally  included  the  cipher,  but  is  now  generally  restricted 
to  i\xQ  first  nine. 

3.  The  last  one  is  called  naught ;  because,  when  standing  alone,  it 
has  no  value,  and  when  connected  with  significant  figures,  it  denotes 
the  absence  of  the  order  in  whose  place  it  stands. 

4.  Zero  is  an  Italian  word,  signifying  nothing. 

The  term  cipher  is  from  the  Arabic  sifr  or  sifreen,  empty,  vacant. 
Subsequently  the  term  was  applied  to  all  the  Arabic  figures  indis, 
crimiuately ;  hence,  calculations  by  them  were  called  ciphering. 

8.  Each  of  the  first  nine  numbers  is  expressed  by  a 
single  figure, — each  figure  denoting  the  number  indicated 
by  its  name.  These  numbers  are  called  units  of  the  first 
order,  or  simply  units. 

8,  a,  Nine  is  the  greatest  tiumber  expressed  by  one  figure. 
Numbers  larger  than  nine  are  expressed  thus : 

Te7i  (i  more  than  9)  is  expressed  by  an  ingenious  de- 
vice, which  groups  ten  single  things  or  ones  together,  and 
considers  the  collection  a  7iew  or  second  order  of  units, 
called  ten.  Hence,  ten  is  expressed  by  writing  the  figure  i 
in  the  second  place  with  a  ciplier  on  the  right;  as,  10. 

The  numbers  from  ten  to  nineteen  inclusive  are  ex- 
pressed by  writing   i  in  the  second  place,  and  the  figure 
denoting  the  units  in  the  first ;  as, 
II,      12,       13,       14,       15,      16,        17,         18,        19. 

eleven,  twelve,  thirteen,  fourteen,  fifteen,  sixteen,  seventeen,  eighteen,  nineteen. 

Twenty  (2  tens)  is  expressed  by  writing  the  figure  2  in 
the  second  place  with  a  ciplier  on  the  right;  as,  20. 

TJiirty  (3  tens)  by  writing  3  in  the  second  place  with  a 
cipher  on  the  right ;  and  so  on  to  ninety  inclusive ;  as, 

20,        30,       40,         50,       60,         70,        80,        90. 

twenty,      thirty,       forty,  fifty,         sixty,      seventy,     eighty,     ninety. 

8.  How  are  the  first  nine  numbers  expressed  ?  What  are  they  called  ?  What 
is  the  greatest  number  expressed  by  one  figure  ?  How  is  ten  expressed  ?  Twenty  ? 
Thirty,  etc.  ?    The  numbers  between  lo  and  20  ?    From  20  to  99  inclusive  ? 


12  l^OTATION". 

The  numbers  from  twenty  to  thirty,  and  so  on  to  ninety^ 
nine  (99)  inclusive,  are  expressed  by  writing  the  tens  in  the 
second  place,  and  the  units  in  the  first ;  as, 


21, 

2  2, 

23, 

34, 

35, 

46, 

57, 

99. 

twenty- 
one. 

twenty- 
two, 

twenty- 
three. 

thirty- 
four, 

thirty- 
five, 

forty- 
six, 

fifty- 
seven, 

ninety- 
nine. 

8,  h.  Ninety-nine  is  the  greatest  number  that  can  be 
expressed  by  tivo  figures. 

A  hundred  (i  more  than  99)  is  expressed  by  grouping 
ten  units  of  the  second  order  together,  and  forming  a  new 
or  third  order  of  units,  called  a  hundred.  Thus,  a  hun- 
dred is  expressed  by  writing  1  in  the  third  place  with  two 
ciphers  on  the  right;  as,  100. 

In  like  manner,  the  numbers  from  one  hundred  to  nine 
hundred  and  ninety-nine  inclusive,  are  expressed  by  writ- 
ing the  hundreds  in  the  third  place,  the  tens  in  the  second, 
and  the  units  in  the  ^r5^.  Thus,  one  hundred  and  thirty- 
five  (i  hundred,  3  tens,  and  5  units,)  is  expressed  by  135. 

8,  c,  Nine  hundred  and  ninety-nine  is  the 

greatest  number  that  can  be  expressed  by  three  figures. 

TJiousands,  and  larger  numbers,  are  expressed  by  form- 
ing other  netv  orders,  called  the  fourth,  fifth,  etc.,  orders ; 
as,  tens  of  thousands,  hundreds  of  thousands,  millions,  etc. 

Notes. — i.  The  names  of  tlie  first  ten  numbers,  one,  two,  tliree, 
etc.,  are  primitive  words.  The  terms  eleven  and  ticelve  are  from  the 
Saxon  endlefen  and  twelif,  meaning  one  and  ten,  two  and  ten.  Thir- 
teen is  from  thir  and  teen,  which  mean  three  and  ten,  and  so  on, 

2.  Twenty  is  from  the  Saxon  tweentig,  tween,  two,  and  ty,  tens; 
i.  e.,  two  tens.     Thirty  is  from  thir  and  ty,  three  tens,  and  so  on. 

3.  The  terms  hundred,  thousand,  and  million  are  primitive  words, 
having  no  perceptible  analogy  to  the  numbers  they  express.- 

From  the  foregoing  illustrations  we  derive  the  following  principle : 
8.  How  is  a  hundred  expresped  ?    ThousandB,  and  larger  numbers  ? 


KOTATION".  It 

9.  The  Orders  of  Units  increase  hy  the  scale  of  ten 
That  is,  ten  single  units  are  one  teri ;  ten  tens  one  liun 
dred ;  ten  hundreds  one  thousand ;  and,  universally. 

Ten  of  any  lower  order  make  a  unit  of  the  nexi 
higher. 

Notes. — i.  If  the  term  unit  denotes  one,  how,  it  may  be  asked,  can 
ten  things  or  ones  be  a  unit,  ten  tens  another  unit,  etc.  And  how 
can  the  figures  i,  2,  3,  etc.,  sometimes  denote  single  things  or  ones  ; 
at  others,  tens  of  ones,  and  so  on. 

The  answer  is,  units  are  of  two  kinds,  simple  and  collective. 

A  simple  unit  is  a  single  thing  or  one. 

A  collective  unit  denotes  a  group  of  ones,  regarded  as  a  whole. 

2.  To  ilfustrate  these  units,  suppose  a  basket  of  pebbles  is  before 
us.     Counting  them  out  one  by  one,  each  pebble  is  a  simple  unit. 

Again,  counting  out  ten  single  pebbles  and  putting  them  togethei 
ill  a  group,  tliis  group  forms  a  unit  of  the  second  order,  called  ten. 
Counting  out  ten  such  groups,  and  putting  them  together  in  one 
pile,  this  collection  forms  a  unit  of  the  third  order,  called  hundred. 
In  like  manner,  a  group  of  ten  hundreds  forms  a  unit  of  the  fourth 
order,  called  thousand,  and  so  on. 

These  different  ^ww^s  of  ten  are  called  units  on  the  same  principle 
that  a  group  of  ten  cents  forms  a  unit,  called  a  dime  ;  or,  a  group  of 
ten  dimes,  a  unit,  called  a  dollar.  Thus,  it  will  be  seen  that  the 
figures  I,  2,  3,  4,  etc.,  always  mean  one,  two,  three,  four  units 
as  their  name  indicates ;  but  the  value  of  these  units  depends  upon 
the  place  the  figure  occupies. 

10.  From  the  preceding  illustrations,  it  will  be  seen 
that  the  Aratic  Notation  is  founded  upon  the  following 
principles : 

ist.  Numbers  are  divided  into  groups  called  units,  of 
theirs/,  second,  third,  etc.,  orders. 

2d.  To  express  these  different  orders  of  units,  a  simple 
and  a  local  value  are  assigned  to  the  significant  figures, 
according  to  the  place  they  occupy. 

3d.  If  any  order  is  wanting,  its  place  is  supplied  by  3. 
cipher. 

9.  How  do  the  orders  of  units  increase?  Units  make  a  ten  ?  Tens  a  hundred* 
10.  Name  the  principles  upon  which  the  Arabic  Notation  is  founded  ? 


14  KOTATIOK. 

11.  The  Simple  Value  of  a  figure  is  the  number 
of  units  it  expresses  when  it  stands  alone,  or  in  the  riglit- 
hand  place. 

The  Local  Value  is  the  number  it  expresses  when 
connected  with  other  figures,  and  is  determined  by  the 
jJilace  it  occupies,  counting  from  the  riqlit. 

12.  It  is  a  general  law  of  the  Arabic  Notation  that  the 
value  of  a  figure  is  increased  tenfold  for  every  place  it  is 
moved  from  the  riglit  to  the  left ;  and,  conversely, 

The  value  of  each  figure  is  diminished  tenfold  for  every 
place  it  is  moved  from  the  left  to  the  right.  Thus,  2  in 
the  first  place  denotes  two  simple  units ;  in  the  second 
place,  ten  times  two,  or  twenty ;  in  the  third  place,  ten 
times  as  much  as  in  the  second  place,  or  two  hundreds, 
and  so  on.     (Art.  8.) 

13.  The  number  denoting  the  scale  by  which  the  orders 
of  units  increase  is  called  the  radix. 

The  radix  of  the  Arabic  Notation  is  ^e^ ;  hence  it  is 
often  called  the  decimal  notation. 

KoTES. — I.  The  term  radix,  Latin,  signifies  root,  or  tase.  The 
term  decimal  is  from  the  Latin  decern,  ten. 

2.  The  decimal  radix  was  doubtless  suggested  by  the  number  of 
fingers  (digiti)  on  both  hands.     (Art.  7,  Note.)     Hence, 

1^.  To  Express  Numbers  by  Figures, 

Begin  at  the  left  hand  and  write  the  figures  of  the  give?i 
orders  in  the  successive  places  toward  the  right. 

If  any  intermediate  orders  are  omitted,  supijly  their 
places  with  ciphers. 


II.  The  simple  value  of  a  figure  ?  Local  ?  12.  What  is  the  law  aa  to  movWg 
a  figure  to  the  right  or  left?  13.  What  is  the  radix  of  a  system  of  notation?  Tho 
radix  of  the  Arabic  ?  What  else  is  the  Arabic  system  called  ?  Why  ?  NoU. 
Meaning  of  radix?  Decimal?  What  Bu^refted  the  decimal  radix?  14.  Rule 
for  expressing  numbers  by  figures  ? 


l^^OTATIOiT. 


EXERCISES. 


Express  Uie  following  numbers  by  figures: 

1.  Three  hundred  and  forty-five. 

2.  Four  hundred  and  sixty. 

3.  Eight  hundred  and  four. 

4.  Two  thousand  three  hundred  and  ten. 

5.  Thirty  thousand  and  nineteen. 

6.  Sixty- three  thousand  and  two  hundred. 

7.  One  hundred  and  ten  thousand  two  hundred  and 
twelve. 

8.  Four  hundred  and   sixty  thousand  nine  hundred 
and  thirty. 

9.  Six  hundred  and  five  thousand  eight  hundred  and 
forty-two. 

10.  Two  millions  sixty  thousand  and  seventy-five. 

ROMAN    NOTATION. 

15.  The  Mofnan  dotation  is  the  method  of  ex- 
pressing numbers  by  certain  letters.  It  is  so  called  because 
it  was  employed  by  the  Romans.  The  letters  used  are 
the  following  seven,  viz. :  I,  V,  X,  L,  C,  D  and  M.  The 
letter  I  denotes  one;  Y,five;  lL,ten;  lu,  fifty ;  Q,  one 
hundred;  J),  five  hundred;  M,  one  thousand.  Interven- 
ing and  larger  numbers  are  expressed  by  the  repetition 
and  combination  of  these  letters. 

16.  The  Eoman  system  is  based  upon  the  following 
general  principles : 

ist.  It  proceeds  according  to  the  scale  of  ten  as  far 
as  a  thousand,  the  unit  of  each  order  being  denoted  by 

15.  Roman  notation?  Why  so  called ?  Letters  employed?  The  letter  I  de- 
note? V?  X?  L?  C?  D?  M?  16.  Name  the  first  principle  upon  which  it 
Is  based.  What  is  the  effect  of  repeating  a  letter  ?  Of  placing  a  letter  of  less 
ralne  before  one  of  greater  value?  If  placed  after?  If  a  line  is  placed  over  a 
tetter? 


16 


N^OTATIOJs". 


a  single  letter.  Thus,  I  denotes  one ;  X,  ten;  C,  one  hun- 
dred ;  M,  one  thousand. 

2d.  Repeating  a  letter,  repeats  its  value.  Thus,  I  denotes 
one ;  II,  two ;  III,  three ;  X,  ten ;  XX,  twenty,  etc. 

3d.  Placing  a  letter  of  less  value  before  one  of  greater 
value,  diminishes  the  value  of  the  greater  by  that  of  the 
less ;  placing  the  less  after  the  greater,  increases  the  value 
of  the  greater  by  that  of  the  less.  Thus,  Y  denotes  five, 
but  IV  denotes  only  four,  and  VI  six. 

4th.  Placing  a  horizontal  line  over  a  letter  increases  its 
value  a  thousand  times.  Thus,  I  denotes  a  thousand  ;  X, 
ten  thousand ;  C,  a  hundred  thousand ;  M,  a  million. 


TABLE. 

I, 

denotes  one. 

XXX, 

denotes  thirty. 

n, 

" 

two. 

XT., 

forty. 

III, 

" 

three. 

L, 

fifty. 

IV, 

" 

four. 

LX, 

sixty. 

V, 

« 

five. 

LXX, 

seventy. 

VI, 

" 

six. 

LXXX, 

eighty. 

VII, 

" 

seven. 

xc. 

ninety. 

VIII, 

" 

eight. 

c, 

one  hundred. 

IX, 

" 

nine. 

oc, 

two  hundred. 

X, 

i> 

ten. 

ccc. 

three  hundred. 

XI, 

" 

eleven. 

cccc, 

four  hundred. 

XII, 

" 

twelve. 

D, 

five  hundred. 

XIII, 

« 

thirteen. 

DC, 

six  hundred. 

XIV, 

" 

fourteen. 

Dec, 

seven  hundred. 

XV, 

(( 

fifteen. 

DCCC, 

eight  hundred. 

XVI, 

" 

sixteen. 

DCCCC 

nine  hundred. 

XVII, 

" 

seventeen. 

M, 

one  thousand. 

XVIII, 

« 

eighteen. 

MM, 

two  thousand. 

XIX, 

" 

nineteen, 

MDCCCLXXI, 

denotes  one  thous- 

XX, 

it 

twenty. 

and  eight  hundred  and  seventy-one. 

Notes. — i.  Four  was  formerly  denoted  by  IIII ;  nine  by  Villi ; 
forty  by  XXXX ;  ninety  by  LXXXX ;  jive  hundred  by  10 ;  and  a 
thousand  by  CIO. 


KOTATIOJ^.  17 

2.  Annexing  0  to  10  (five  hundred)  increases  its  value  ten  times. 
Thus,  loo  denotes  five  thousand ;  lOOO,  fifty  thousand. 

Prefixing-  a  C  and  annexing  a  0  to  CIO  (a  thousand)  increases  its 
value  te7i  times.     Thus,  CCIOO  denotes  ten  thousand,  etc.    Hence, 

17.  To  Express  Numbers  by  Letters, 

Begin  at  the  left  hand  or  highest  order,  and  write  the 
letters  denoting  the  given  number  of  each  order  in  successioti. 

Note. — The  Roman  Notation  is  seldom  used,  except  in  denoting 
chapters,  sections,  heads  of  discourses,  etc. 

EXERCISES. 

Express  the  following  numbers  by  letters  : 

I.  Fourteen,  2.  Twenty-nine,  3.  Thirty-four, 

4.  Sixty-six,  5.  Forty-nine,  6.  Seventy-three, 

7.  Eighty-eight,        8.  Ninety-four,  9.  Ninety-nine, 

10.  107,  II.  212,  12.  498, 

13.  613,  14.  507,  15.  608. 

16.  724,  17.  829,  18.  928, 

19.  1004,  20.  1209,  21.  1363, 

22.  1417,  23.  1614,  24.  1671, 

25.  1748,  26.  1803,  27.  1876. 


NUMERATION. 

18.  Numeration  is  the  art  of  reading  numbers  ex- 
pressed by  figures,  letters,  or  other  numeral  characters. 

Note. — The  learner  should  be  careful  not  to  confound  JV^umeration 
with  Notation.  The  distinction  between  them  is  the  same  as  that 
between  reading  and  writing. 

19.  There  are  two  methods  of  reading  numbers,  the 
French  and  the  English. 

17.  Rule  for  expressing  numbers  by  letters?  18.  Numeration?  Note.  Dis- 
tinction between  Numeration  and  Notation  ?  19.  How  many  methods  of  reading 
numbers  ? 


18  NUMERATIOJS". 


FRENCH    NUMERATION. 

20.  The  French  divide  numbers  into  periods  of  thre^ 
■figures  each,  and  then  subdivide  each  period  into  units, 
tens,  and  hundreds,  as  in  the  following 

TABLE. 


o  .  tj 

SS  ^S  ;=1«^  :3»3  _2g 

■53      h|      s§      s|      h| 

■ol  2     -Sig         "sa         =53         -Sj  ^       . 

rS^-^'^  nj'^S  r^W.  •73'^<v:  'C'^g  'O 

Q3V-:§  SvhSS  ®Cf-,S  f'ttff  f'S*  9 

flsl    Isi    |§i    Isi    gsi   Is I 
Sl&   ISS    w^§    1^^    l^g    B^g 

823        561        729        452        789        384 
6th  period.  5th  period.  4th  period.    3d  period.     2d  period.    1st  period. 

The  first  period  on  the  right  is  called  units  period ;  the 
second,  thousands ;  the  third,  millions,  etc. 

The  periods  in  the  table  are  thus  read  :  823  quadrillions, 
561  trillions,  729  billions,  452  millions,  789  thousand,  three 
hundred  and  eighty-four. 

Note. — The  terms  billion,  trillion,  quadrillion,  etc.,  are  derived 
from  the  Italian  milione  and  the  Latin  bis,  tres,  quatuor,  etc.  Thus, 
bis,  united  with  million,  beccjiies  billion,  etc. 

21.  To  read  Numbers  according  to  the  French   Numeration. 

Divide  them  into  periods  of  three  figures  each,  counting 
from  the  right. 

Beginning  at  the  left  hand,  read  the  periods  in  succes- 
sion, and  add  the  name  to  each,  except  the  last. 

20.  The  French  method  ?  Repeat  the  Table,  beginning  at  the  right.  What  is 
the  1 6t  period  called?  The  2d?  3d?  4th?  5th?  6th?  21.  The  rule  for  reading 
numbers  by  the  French  method  ?  Note.  Why  omit  the  name  of  the  right  hand 
period  ?    The  difference  between  orders  and  periods  ? 


IfUMERATIOir. 


19 


NoTBS. — I.  The  name  of  the  right-hand  period  is  omitted  for  the 
of  conciseness  ;  and  since  this  period  always  denotes  units,  the 
omission  occasions  no  obscurity. 

2.  The  learner  should  observe  the  diflerence  between  the  orders  of 
units  and  the  periods  into  which  they  are  divided.  Tlie  former 
increase  by  tens  ;  the  latter  by  thousands. 

3.  This  method  of  reading  numbers  is  commonly  ascribed  to  the 
French,  and  is  thence  called  French  numeration.  Others  ascribe  it  to 
the  Italians,  and  thence  call  it  the  Italian  method. 


Read  the  following  numbers  by  the  French  riumeratipn  : 


I. 

270 

II. 

840230 

21.     3006017 

2. 

309 

12. 

4603400 

22.  2460317239 

3- 

1270 

13-    35040026 

23.  5100024000 

4. 

2036 

14.    80600000 

24.    70300510 

5- 

8605 

15- 

9001307 

25.   203019060 

6. 

40300 

16.    65023009 

26.  7800580019 

7. 

85017 

17. 

810000 

27.  86020005200 

8. 

160401 

18.    75306020 

28.    51036040 

9- 

405869 

19.   165380254 

29.   621000031 

10. 

1365406 

20.   31 

0400270 

30.  93000275320 

31- 

216 

327250516 

3^' 

289300210375861 

32. 

4260 

300210109 

37. 

5400000541000600 

33- 

300 

073004000 

38. 

60005 10000243000 

34. 

41295 

000400649 

39- 

40200000008060704 

35- 

264300 

439000200 

40.    6 

00040300607230516 

Express  the  following  numbers  by  figures 

1.  Ten  millions,  five  thousand,  and  two  hundred. 

2.  Sixty-one  millions,  three  hundred  and  forty. 

3.  Three  hundred  and  ten  millions,  and  five  hundred. 

4.  Twenty-six  billions,  seventy  milHons,  three  hundred. 

5.  One  hundred  billions,  four  hundred  and  twenty-five. 

6.  Sixty-eight  trillions,  seven  hundred  and  twenty-five. 

7.  Eight  hundred  and  twenty  millions,  five  hundred 
and  twenty-three. 


20  NUMEEATIOK. 

8.  Sixty-seven  quadrillions,  ninety-seven  billions. 

9.  Four  hundred  and  sixty  quadrillions,  and  eighty- 
seven  millions. 

10.  Seven  hundred  and  sixty-one  quadrillions,  seventy- 
one  trillions,  two  hundred  billions,  eighteen  millions,  five 
thousand,  and  thirty-six. 

ENGLISH    NUMERATION. 

22.  The  English  divide  numbers  into  periods  of  six 
•figures  each,  and  then  subdivide  each  period  into  units, 
tens,  hundreds,  thousands,  tens  of  thousands,  and  hundreds 
of  thousands,  as  in  the  following 

TABLE. 


o 
m  -A 


-     1 «  fl 


^'  I  «  S  §        ^-  i  ^  §  i  e  i 

'2o2f^'Sa2.S            'SoQ^'^mJ  'SoDPi'^aQ-S 

QdogS;:§          SfloSfl!^  SaoSS'g 

EdS^pg^          s^Srdf^Sts  f^SpdJ^Si^ 

HHHW^ft^        WhSw^J^  WHHWHti 

407692         958604  413056 

V ^ .  V ^ /  V ^ . 

3d  period.                        2d  period,  ist  period. 

The  periods  in  the  Table  are  thus  read :  407692  billions,  958604 
millions,  413  thousand,  and  fifty-six. 

Note. — This  method  is  called  English  numeration,  because  it  was 
invented  by  the  English. 

For  other  numbers  to  read  by  this  method,  the  pupil  is  referred 
to  those  in  Art.  21. 

aa.  What  is  the  English  method  of  numeration  ?  Note.  Why  so  called  ? 


ADDITION. 

23.  Addition  is  uniting  two  or  more  numbers  in  ona 
The  Sum  or  Afuount  is  the  number  found  by  addi- 
tion.    Thus,  5   added  to  7  are  12;   twelve,  the  number 
obtained,  is  the  sum  or  amount. 

Notes. — i.  The  sum  or  amount  contains  as  many  units  as  tlio 
numbers  added.  For,  the  numbers  added  are  composed  of  units ; 
and  the  whole  is  equal  to  the  sum  or  all  its  parts.    (Art.  3.) 

2.  When  the  numbers  added  are  the  same  denomination,  the 
operation  is  called  Simple  Addition. 

SIGNS. 

24.  Signs  are  characters  used  to  indicate  the  relation 
of  numbers,  and  operations  to  be  performed. 

25.  The  Sign  of  Addition  is  a  perpendicular 
cross  called  plus  (  +  ),  placed  before  the  number  to  be 
added.  Thus  7+5,  means  that  5  is  to  be  added  to  7,  and 
is  read  "  7  plus  5." 

Note. — The  term  plus,  Latin,  signifies  inore,  or  added  to. 

26.  The  Sign  of  JEquality  is  two  short  parallel 
lines  (  =  ),  placed  between  the  numbers  compared.  Thus 
7+5  =  12,  means  that  7  and  5  are  equal  to  12,  and  is  read, 
"  7  plus  5  equal  12,"  or  the  sum  of  7  plus  5  equals  12. 

Eead  the  following  numbers : 

1.  8  +  4  +  2  =  6  +  8  4.  23+   7  =  19  +  11 

2.  7+3  +  5  =  2  +  1  +  12  5.  37+   8=30  +  15 

3.  19  +  1+0=6  +  5+    9  6.  58  +  10  =  40  +  28 


23.  What  is  addition?  The  result  called?  Note.  When  the  numbers  added 
are  the  same  denomination,  what  is  the  operation  called  ?  24.  What  are  signs  ? 
25.  The  sign  of  addition ?  Note.  The  meaning  ot plus?  26.  Sign  of  equality? 
How  is  7+5=12  read? 


22 


ADDITIOK. 


27.  The  Sign  of  Dollars  is  a  capital  S  with  two 
perpendicular  marks  across  it  ($),  prefixed  to  the  number 
of  dollars  to  be  expressed.     Thus,  I245  means  245  dollars. 

Note. — The  term  prefix,  from  the  Latin  prefigo^  signifies  to  pla<;e 
before. 
Bead  the  following  expressions : 

1.  $174-    $8=|i5+|io      I      4.     $25+    |8=:|lO  +  $23 

2.  $i3  +  $2o  =  |i6+$i7  5-     l25+$4o=l5o-|-$i5 

3.  $21+   $7  =  $i2+$i6      1     6.  $1054- 136  — $96  +  145 

ADDITION  TABLE. 


I 

and 

2 

and 

3 

an 

d 

4 

and 

5 

and 

I 

are  2 

I 

are  3 

I 

are 

4 

I 

are  5 

I 

are  6 

2 

3 
4 
5 
6 

"   3 
"  4 

"   5 
"   6 

"   7 

2 

3 

4 

5 
6 

"   4 

"   5 
"   6 

"   7 
«   8 

2 

3 

4 

5 
6 

a 

(( 

5 
6 

7 
8 

9 

2 

3 

4 

5 
6 

"  6 

"   7 
"   8 

"   9 

«  xo 

2 
3 

4 

5 
6 

"  7 
"   8 

"  9 
"  10 
«  II 

7 
8 

"   8 
"   Q 

7 
8 

"   9 
"  10 

7 
8 

10 
II 

7 
8 

"  II 
"  12 

7 
8 

"  12 
"  13 

9 

"  10 

9 

"  II 

9 

a 

12 

9 

"  13 

9 

"  14 

10 

"  II 

10 

"  12 

10 

(( 

13 

10 

"  14 

10 

"  15 

6  and 

7 

and 

8  and 

9 

and 

10  and 

I 

are  7 

I 

are  8 

I 

are 

9 

I 

are  10 

I 

are  11 

2 

"  8 

2 

"   9 

2 

a 

10 

2 

"  II 

2 

"  12 

3 

4 

5 
6 

"   9 
«  10 
"  II 
"  12 

3 

4 

5 
6 

"  10 
"  II 
"  12 
"  13 

3 
4 

5 
6 

(( 
a 

a 
a 

II 
12 
13 
14 

3 

4 

5 
6 

"  12 
"  13 
"  14 
"  IS 

3 
4 

5 
6 

"  13 

"  14 

"  15 
"  16 

7 
8 

"  13 

"  14 

7 
8 

"  14 

"  IS 

7 
8 

15 
16 

7 
8 

"  16 

"  17 

7 
8 

"  17 
"  18 

9 
10 

"  15 
"  16 

9 
10 

"  16 
"  17 

9 
10 

17 

18 

9 
10 

«  18 
"  19 

9 
10 

"  19 

"  20 

^^  More  mistakes  are  made  in  adding  than  in  any  other  arith- 
metical operation.  The  first  five  digits  are  easily  combined ;  the 
results  of  addin)^  g,  being  i  less  than  if  10  were  added,  are  also  easy. 
The  others,  6,  7,  8,  are  more  difficult,  and  therefore  should  receive 
sveciaZ  attention. 


25.  What  is  the  sign  of  dollars? 


ADDITION.  23 

CASE    I. 

28.   To  find  the  A^nount  of  two  or  more  numbers,  when 
the  Sum  of  each  column  is  Less  than  10. 

Ux.  I.  A  man  owns  3  farms;  one  contains  223  acres, 
another  51   acres,  and  the  other  312  acres:   how  many 
acres  has  he  ? 
Analysis. — Let  the  numbers  he  set  down  as  in 

r,       •         •  -,      ■  -,  OPERATION. 

the  margin.     Beginning  at  the  right,  we  proceed  ^• 

thus:  2  units  and  i  unit  are  3  units,  and  3  are  6  o  S-^ 

units ;  the  sum  being  less  than  ten  units,  we  set  it  aS  ^ 

under  the  column  of  units,  because  it  is  units.    Next,  ^  ^3 
I  ten  and  5  tens  are  6  tens,  and  2  are  8  tens ;  the  5 1 

sum  being  less  than  10  tens,  we  set  it  under  the  ■?  1 2 

column  of  tens,  because  it  is  tens.     Finally,  3  hun-  

dreds  and   2  hundreds  are  5  hundreds;   the  sum     Ans.  586 
being  less  than  10  hundreds,  we  set  it  under  the 

column  of  hundreds,  for  the  same  reason.     Therefore,  he  has  586 
acres.    All  similar  examples  are  solved  in  like  manner. 

By  inspecting  the  preceding  illustration,  the  learner 
will  discover  the  following  principle : 

Units  of  the  same  order  are  added  together,  and  the 
sum  is  placed  under  the  column  added.     (Art.  9.) 

Notes. — i.  The  same  orders  are  placed  under  each  other  for  the 
sake  of  convenience  and  rapidity  in  adding. 

2.  We  add  the  same  orders  together,  units  to  units,  tens  to  tens, 
etc.,  because  different  orders  express  units  of  different  values,  and 
therefore  cannot  be  added  to  each  other.  Thus,  5  units  and  5  tens 
neither  make  10  units  nor  10  tens,  any  more  than  5  cents  and  5 
dimes  will  make  10  cents  or  10  dimes. 

3.  We  add  the  columns  separately,  because  it  is  easier  to  add  one 
order  at  a  time  than  several. 

4.  The  sum  of  each  column  is  set  imder  the  column  added,  because 
being  less  than  10,  it  is  the  .mme  order  as  that  column. 

(2.)  (3.)  (4.)  (S-)  (6-) 

2102  21032 
1253  52010 
4604        24603 


2103 

3024 

I2II 

4022 

1230 

2002 

1674 

4603 

5340 

7.  Wliat  is  the  sum  of  $2321 +$123 +  $3245  ? 


2i  ADDITIOir. 

8.  What  is  the  sum  of  3210  pounds  +  2023  pounds  + 
4601  pounds? 

9,  What  is  the  sum  of  130230  +  201321+402126  ? 
10.  What  is  the  sum  of  2410632  + 1034246 +  3201 20  ? 

CASE    II. 

29.  To  find  the  Amount  of  two  or  more  numbers,  when 

the  Sum  of  any  column  is  10,  or  more. 

I.  What  is  the  sum  of  $436,  I324,  and  $645  ? 

Analysis. — Let  the  numbers  be  set  down  as  in  the 
margin.     Adding  as  before,  the  sum  of  the  first  column      operation 
is  15  units,  or  i  ten  and  5  units.     We  set  the  5  units        ^43^ 
under  the  column  added,  and  add  the  i  ten  to  the  next  324 

column  because  it  is  the  same  order  as  that  column.  645 

Now,  I  added  to  4  tens  makes  5  tens,  and  2  are  7  tens,  and        

3  are  10  tens,  or  i  hundred  and  o  tens.  We  set  the  o,  or  $1405 
right  hand  figure,  under  the  column  added,  and  add  the 
I  hundred  to  the  next  column,  as  before.  The  sum  of  the  next 
column,  with  the  i  added,  is  14  hundreds  ;  or  i  thousand  and  4  hun- 
dreds. This  being  the  last  column,  we  set  down  the  wliok  sum. 
The  answer  is  $1405.  All  similar  examples  may  be  solved  in  like 
manner. 

By  inspecting  this  illustration,  it  will  be  seen, 

When  the  sum  of  a  column  is  10  or  more,  we  write  the 
uiiits'  figure  under  the  column,  and  add  the  tens'  figure  to 
the  next  column. 

Notes. — i.  We  set  the  units^  figure  under  the  column  added,  and 
add  the  tens  to  the  next  column,  because  they  are  the  same  orders 
as  these  columns. 

2.  We  begin  to  add  at  the  right  hand,  in  order  to  carry  the  tens  as 
we  proceed.  We  set  down  the  tchole  sum,  of  the  last  column,  because 
there  are  no-  fig-ures  of  the  same  order  to  which  its  left  hand  figure 
can  be  added. 

30.  Adding  the  tens  or  left  hand  figure  to  the  next 
column,  is  called  carrying  the  tens.  The  process  of  carry- 
ing the  tens,  it  will  be  observed,  is  simply  taking  a  certain 
number  of  units  from  a  lower  order,  and  adding  their 
equal  to  the  next  higher;  therefore,  it  can  neither  increase 
nor  diminish  the  amount. 


ADDITION.  25 

Note. — We  carry  for  ten  instead  of  seven,  nine,  eleven,  etc., 
because  in  the  Arabic  notation  the  orders  increase  by  the  scale  of 
t^.n.  If  they  increased  by  the  scale  of  eight,  twelve,  etc.,  we  should 
carry  for  that  number.    (Art.  13.) 

(2.)  (3.)  (4.)  (5.)  (6.) 

5689  6898  7585  8456  97504 

3792  3365  3748  5078  s^y86 

4358  79S7  8667  6904  75979 

/3T57      /flTo     3uToo      si^t     aJsJUJ 

31.  THe  preceding  principles  may  be  summed  up  in 
the  following 

GENERAL   RULE. 

I.  Place  the  numhers  one  under  another,  units  under 
units,  etc. ;  and  beginning  at  the  right,  add  each  column 
separately. 

II.  If  the  sum  of  a  column  does  not  exceed  nine,  ivrite 
it  under  the  column  added. 

If  the  sum  exceeds  nine,  write  the  units'^  figure  under 
the  column,  and  add  the  tens  to  the  next  higher  order. 
Fi7ially,  set  down  the  whole  sum  of  the  last  column. 

Notes. — i.  As  soon  as  the  pupil  understands  the  principle  of 
adding,  he  should  learn  to  abbreviate  the  process  by  simply  pro- 
nouncing the  successive  results,  as  he  points  to  each  figure  added. 
Thus,  instead  of  saying  7  units  and  9  units  are  16  units,  and  8  are 
24  units,  and  7  are  31  units,  he  should  say,  nine,  sixteen,  twenty-four, 
tliirty-one,  etc. 

Again,  if  two  or  more  numbers  together  make  10,  as  6  and  4, 
7  and  3  ;  or  2,  3,  and  5,  etc.,  it  is  shorter,  and  therefore  better,  to  add 
10  at  once. 


31.  How  write  numbers  to  be  added  ?  The  next  step  ?  If  the  sum  of  a  column 
does  not  exceed  nine,  what  do  you  do  with  it?  If  it  exceeds  nine?  The  sum 
of  the  last  column  ?  28.  Note.  Why  write  units  under  units,  etc.  ?  Why  add  the 
columns  separately  ?  Why  not  add  diflferent  orders  together  promiscuously  ?  Is 
the  sum  of  3  units  and  4  tens,  7  units  or  7  tens  ?  When  the  sum  of  a  column  does 
not  exceed  9,  why  set  it  under  the  column  ?  29.  Note.  If  the  sum  of  a  column  is 
10  or  more,  why  set  the  units'  figure  under  the  column  added,  and  carry  the  tens 
to  the  next  column  ?  30.  What  is  meant  by  carrying  the  tens  ?  Why  does  not 
carrying  change  the  amount  ?    Why  carry  for  10  instead  of  6,  8,  12,  etc. 


26  ADDITIOl^. 

2.  Accountants  sometimes  set  the  figure  carried  under  the  right 
hand  figure  in  a  line  below  the  answer.  In  this  way  the  sum  of 
each  column  is  preserved,  and  any  part  of  the  work  can  be  reviewed, 
if  desired  ;  or  if  interrupted,  can  be  resumed  at  pleasure. 

32.  V'^00'^.— Begin  at  the  top  and  add  each  column 
dowmvard.    If  the  two  results  agree,  the  work  is  right. 

Note. — This  proof  depends  upon  the  supposition  that  reversing 
the  order  of  the  figures,  will  detect  any  error  that  may  have  occurred 
in  the  operation. 

EXAMPLES. 

I.  Find  the  sum  of  864,  741,  375,  284,  and  542,  and 
prove  the  operation.    .*  S^  " 


(^•) 

(3.) 

(4.) 

(5.) 

(6.) 

Dollars. 

Pounds. 

Yards. 

Rods. 

Feet. 

263 

4780 

2896 

23721 

845235 

425 

7642 

8342 

70253 

476234 

846 

5036 

257 

4621 

6897 

407 

7827 

s  the  sun 

3261 

342 

723468 

7.  What  i 

1  of  675  acres +  84i  acr^  +  9'04  acres 

4-39  acres? 

8.  What 

is  the  sum  of  8423  +  286  +  793: 

2  +  28  +  6790? 

9.  What 

is  the  sum  of  824^ 

11+376  +  19 

1  +  62328  +  4521 

+  35787? 

10.  What 

is  the  sum  of  63^ 

.28  +  78  +  4236  +  628  +  93  + 

S413? 

(II.) 

(12.) 

(13.) 

(14.) 

(15.) 

2685 

89243 

72094 

825276 

9031253 

6543 

8284 

96308 

704394 

432567 

8479 

34567 

763 

37783 

65414 

6503 

865 

4292 

1697 

9236 

1762 

3952 

23648 

349435 

843 

7395 

42678 

75965         697678 
$2358  for  his  farm,  $ 

68 

J[6.  If  a  man  pays 

1950  for  stock, 

end  I360  for  tools,  how  much  does  he  pay  : 

for  all? 

32.  How  prove  addition  ?    Not-e.  Upon  what  is  this  proof  based  ? 


ADDITION.  27 

17.  A  merchant  bought  371  yards  of  silk,  287  yards  of 
calico,  643  yards  of  muslin,  and  75  yards  of  broadcloth: 
how  many  yards  did  he  buy  in  all  ? 

18.  Eequired  the  sum  of  $2404  4- $100 +  $1965 +  $1863. 

19.  Eequired  the  sum  of  968  pounds +  81  pounds +  7 
pounds +  639  pounds. 

20.  Required  the  sum  of  1565  gals. +  870  gals.  +  3i 
gals.  +  160  gals.  +  42  gals. 

21.  Add  2368,  1764,  942,  87,  6,  and  5271. 

22.  Add  281,  6240,  37,  9,  1923,  loi,  and  45. 

23.  Add  888,  9061,  75,  300,  99,  6,  and  243. 

24.  243  -1-765  -1-980  + 759 -f  127  =how  many? 

25.  9423 +  100 -I- 1600 -f- 1 19 +  4004=::  how  many? 

26.  81263  +  16319  +  805  +2500  +  93  — how  many? 

27.  236517+460075  +  235300  + 275i6i  =  how  many? 

28.  A  savings  bank  loaned  to  one  customer  $1560,  to 
another  $1973,  to  a  third  I2500,  and  to  a  fourth  $3160; 
how  much  did  it  loan  to  all  ? 

29.  A  young  farmer  raised  763  bushels  of  wheat  the 
first  year,  849  bushels  the  second,  10 11  bushels  the  third, 
and  1375  bushels  the  fourth:  how  many  bushels  did  he 
raise  in  4  years  ? 

30.  A  man  bequeathed  the  Soldiers'  Home  $8545 ;  the 
Blind  Asylum  I7538;  the  Deaf  and  Dumb  Asylum 
$6280;  the  Orphan  Asylum  $19260;  and  to  his  wife  the 
remainder,  which  was  $65978.  What  was  the  value  of 
his  estate  ? 

31.  A  merchant  owns  a  store  valued  at  $17265,  his 
goods  on  hand  cost  him  $19230,  and  he  has  $1563  in 
bank :  how  much  is  he  worth  ? 

32.  What  is  the  sum  of  thirteen  hundred  and  sixty- 
three,  eighty-seven,  one  thousand  and  ninety-four,  and 
three  hundred  ? 

33.  What  IS  the  sum  of  three  thoueand  two  hundred 
and  forty,  fifteen  hundred  and  sixty,  and  nine  thousand? 


'ZS  ADDITIOi?^. 

34.  Add  ninety  thousand  three  hundred  and  two,  sixty- 
five  thousand  and  thirty,  forty-four  hundred  and  twenty- 
three. 

35.  Add  eight  hundred  thousand  and  eight  hundred, 
forty  thousand  and  forty,  seven  thousand  and  seven,  nine 
hundred  and  nine. 

36.  Add  30006,  301,  55000,  2030,  67,  and  95000. 

37.  Add  65139,  looioo,  39,  1 1 II II,  763002,  and  317. 

38.  Add  81,  907,  311,  685,  9235,  7,  and  259. 

39.  Add  4895,  352,  68,  7,  95,  645,  and  3867. 

40.  Add  631,  17,  I,  45,  9268,  196,  and  3562. 

41.  Add  77777,  3333S,  88888,  22222,  and  iiiii. 

42.  Add  236578,  125,  687256,  and  404505- 

43.  Add  23246,  8200461,  5017,  8264,  and  39. 

44.  Add  317,  21,  9,  4500,  219,  3001,  17036,  and  45. 

45.  Add  1 0000000,  1 000000,  1 00000,  1 0000,  1000,  100, 
tind  10. 

46.  Add  22000000,  22000,  202000,  2200,  and  220. 

47.  If  a  young  man  lays  up  $365  in  i  year,  how  much 
•will  he  lay  up  in  4  years  ? 

48.  In  what  year  will  a  man  who  was  horn  in  1850,  be  75 
years  old  ? 

49.  A  man  deposits  $1365  in  a  bank  per  day  for  6  days 
in  succession:  what  amount  did  he  deposit  during  the 
week? 

50.  A  planter  raised  1739  pounds  of  cotton  on  one 
section  of  his  estate,  703  pounds  on  another,  2015  pounds 
on  another,  and  2530  pounds  on  another:  how  many 
pounds  did  he  raise  in  all  ? 

51.  A  man  was  29  years  old  when  his  eldest  son  was 
born ;  that  son  died  aged  47,  and  the  father  died  1 7  years 
later:  how  old  was  the  man  at  his  death? 

52.  A  speculator  bought  3  city  lots  for  $21213,  ^-nd  sold 
so  as  to  gain  I375  on  each  lot:  for  what  sum  di^  he  sell 
them? 


ADDITION.  29 

53.  A's  income-tax  for  1865  was  $4369,  B's  $3978;  C's 
was  $135  more  than  A's  and  B^s  together,  and  D's  was 
equal  to  all  the  others:  what  was  D's  tax?  Yfhat  the 
tax  of  all  ? 

54.  How  many  strokes  does  a  clock  strike  in  24  hours? 

55.  A  raised  3245  bushels  of  corn,  and  B  raised  723 
bushels  more  than  A :  how  many  bushels  did  both  raise  ? 

56.  In  a  leap  year  7  months  have  31  days  each,  4  months 
30  days  each,  and  i  month  has  29  days:  how  many  days 
constitute  a  leap  year  ? 

57.  The  entire  property  of  a  bankrupt  is  $2648,  which 
is  only  half  of  what  he  owes :  how  much  does  he  owe  ? 

58.  What  number  is  19256  more  than  31273? 

59.  A  has  860  acres  of  knd,  B  117  acres  more  than  A, 
and  G  as  many  as  both  :  how  many  acres  have  all  ? 

DRILL     COLUMNS. 


(60.) 

(61.) 

(62.) 

(63.) 

(64.) 

(65.) 

Pols.  cts. 

Dols.  cts. 

Dols.  cts. 

Dels.  cts. 

Dols.  cts. 

Dols.  cts. 

14  65 

25  76 

37  38 

24  91  - 

34  45 

49  68 

21  43 

32  37 

3  25  . 

42  73 

32  61 

26  52 

64  61 

54  61 

46  72 

9  61 

64  12 

3^   43 

37  28 

16  45 

28  41 

32  44 

70  33 

27  59 

43  24 

67  38 

7  50 

51  62 

24  54 

12  38 

46  03 

34  92 

63  04 

9  28 

32  67 

45  63 

19  41 

76  41 

16  28 

70  s6 

48  32 

50  71 

32  34 

47  69 

7  39 

84  52 

25  67 

26  8s 

34  32 

31  04 

82  01 

19  24 

13  09 

34  26 

15  73 

83  26 

7  63 

32  41 

58  32 

72  61 

62  64 

27  13 

24  07 

42  35 

24  6s 

23  45 

45  76 

74  52 

52  34 

56  72 

72  56 

67  80 

The  ability  to  add  with  rapidity  and  correctness  is  con. 
fessedly  a  most  valuable  attainment ;  yet  it  is  notorious  that  few 
pupils  ever  acquire  it  in  school.  The  cause  is  the  neglect  of  tho 
Table,  and  the  want  of  drilling.    Practice  is  the  price  of  skill. 


30 

ADDITIOi^ 

• 

(66.) 

(67.) 

(68.) 

(69.) 

(70.) 

Dols.  cts. 

Dols. 

cts. 

Dols. 

cts. 

Dole. 

cts. 

Dols.  cts. 

25  63 

346 

25 

273 

42 

3424 

27 

4386  48 

46  74 

405 

31 

534  97 

6213 

39 

3275  26 

82  31 

830 

62 

286 

31 

5382 

13 

5327  64 

60  46 

642 

43 

347 

34 

4561 

63 

4613  04 

75  38 

761 

38 

721 

35 

8324 

29 

1729  56 

22  76 

482 

71 

635 

26 

5276 

34 

3537  63 

64  28 

395 

65 

453 

94 

6594 

32 

8274  31 

37  31 

762 

34 

587 

63 

1723 

45 

5026  73 

25  16 

250 

25 

658 

92 

2674  56 

1586  37 

32  10 

S74  50 

894 

28 

3295 

31 

2345  67 

75  S7 

328 

25 

946 

25 

4463 

79 

4326  43 

62  75 

432 

12 

973 

22 

5324 

28 

5437  26 

48  12 

519 

75 

564 

33 

6543 

20 

6325  41 

23  15 

438 

29 

283 

12 

7035 

28 

7396  27 

18  II 

533 

25 

597 

31 

8546 

37 

8325  34 

50  25 

453 

62 

296 

54 

9634 

25 

9274  52 

78  26 

374 

52 

458 

73 

8346 

52 

3465  23 

39  44 

250 

37 

605 

42 

6275 

35 

4289  67 

72  51 

862 

75 

486 

54 

4235 

34 

5365  72 

33   41 

953 

25 

193 

12 

3271 

05 

6582  39 

58  75 

846 

62 

586 

32 

6137 

94 

1205  32 

29  31 

553 

12 

lOI 

53 

6283 

59 

4396  84 

54  62 

228 

51 

^53 

72 

4346 

32 

5724  33 

27  31 

312 

52 

154 

53 

7294 

58 

1065  43 

29  50 

455 

63 

276 

32 

6275 

63 

4953  27 

68  71 

729 

31 

586 

34 

3284 

32 

2586  54 

97  53 

426  76 

235 

20 

^^35 

34 

4234  62 

32  43 

623 

25 

463 

52 

2586  89 

1736  44 

64  25 

321 

35 

958  76 

7434 

26 

5398  29 

18  12 

238 

17 

386 

29 

5869 

73 

1234  56 

19  50 

125 

51 

3^5 

46 

3276 

42 

7891  01 

62  25 

436 

25 

434 

57 

1635  38 

1234  16 

64  37 

536  63 

372 

46 

5913 

84 

6843  75 

53  63 

257 

47 

657 

32 

6284 

35 

7616  24 

SUBTRACTION. 

33.  Subtrciction  is  taking  one  number  from  another. 
The  Subtrahend  is  the  number  to  be  subtracted. 
The  Minuend  is  the  number  from  which  the  sub- 
traction is  made. 

The  Difference  or  He^nainder  is  the  number  found 
by  subtraction.  Thus,  when  it  is  said,  5  taken  from  1 2 
leaves  7,  12  is  the  minuend;  5  the  subtrahend;  and  7  the 
difference  or  remainder. 

Notes. — i.  Tho  term  subtraction,  is  from  the  Latin  subtraho,  to 
draw  from  under,  or  take  away. 

The  term  subtrahend,  from  the  same  root,  signifies  that  which  is  to 
be  subtracted. 

Minuend,  from  the  Latin  minuo,  to  diminish,  signifies  that  which 
is  to  he  diminished  ;  the  termination  nd  in  each  caso  having  the  force 
of  to  be. 

2.  Subtraction  is  the  opposite  of  Addition.  One  imites,  the  other 
separates  numbers. 

3  When  both  numbers  are  the  same  denomination,  the  operation 
is  called  Siiuple  Subtraction. 

34.  The  Sign  of  Subtraction  is  a  sJiort  horizo7ital 
line  called  fnimis  (  — ),  placed  before  the  number  to  be 
subtracted.  Thus,  6—4  means  that  4  is  to  be  taken  from 
.6,  and  is  read,  "  6  minus  4." 

Note. — The  term  minus,  Latin,  signifies  less,  or  diminished  bj. 
Eead  the  following  expressions : 

I.  21—  3  =  2+   3  +  13.         2.     31  — 103=  6+  4+11. 

3.  45  —  15  =  8  +  12  +  10.         4.  100—30  =  55  +  10+   5. 


-^3.  What  is  subtraction  ?  What  is  the  number  to  be  subtracted  called  ?  The 
nwjibar  fVom  which  the  subtraction  is  made?  The  result?  Note.  Meaninfr  of 
te%,x  Subtraction  ?  Subtrahend  ?  Minuend  ?  Difference  between  Subtraction 
anC  (Addition  ?  When  the  Nos.  are  the  same  denomination,  what  is  the  operation 
called  ?    34.  Si^  of  Subtraction  ?    How  read  6  -4  ?    Meaning  of  the  term  minus : 


32 


SUBTRACTIOiq^. 


SUBTRACTION   TABLE. 


I  from 

2 

from 

3  from 

4  from 

5  from 

1  leaves 

2  " 

o 

I 

2  leaves 

3  " 

o 

I 

3  leaves 

4  " 

o 

I 

4  leaves 

5  " 

o 

I 

5  leaves  o 

6  "       I 

3  " 

4  " 

5  " 

6  " 

7  " 

2 

3 

4 
5 
6 

4 

5 
6 

7 
.8 

a 
(( 
a 

2 

3 

4 

5 
6 

5  " 

6  " 

7  " 

8  « 

9  " 

2 

3 

4 

5 
6 

6  " 

7  " 

8  « 

9  " 

lO      « 

2 

3 

4 

5 
6 

7  "          2 

8  "      3 

9  "      4 

10  "      5 

11  "      6 

8  « 

9  " 

7 
8 

9 

lO 

7 
8 

10  " 

11  " 

7 
8 

11  " 

12  " 

7 
8 

12  "       7 

13  "      8 

lO      « 

9 

II 

a 

9 

12       « 

9 

13     " 

9 

14     "      9 

II     " 

lO 

12 

lO 

13     " 

IC 

14     " 

lO 

15     "     10 

6  from 

7 

from 

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6  leaves 

7  " 

o 

I 

7  leaves 

8  " 

o 

I 

8  leaves 

9  " 

o 

I 

9  leaves 
lO       « 

o 

I 

10  leaves  0 

11  "       I 

8     « 

2 

9 

a 

2 

lO      " 

2 

II      " 

2 

12      "       2 

9     " 

10  " 

11  " 

12  " 

3 
4 

5 
6 

lO 

II 

12 

13 

a 

a 

3 

4 

5 
6 

11  " 

12  « 

13  " 

14  " 

3 

4 

5 
6 

12  " 

13  " 

14  " 

15  " 

3 

4 

5 
6 

13  "      3 

14  "      4 

15  "      5 

16  "      6 

13  " 

14  " 

7 
8 

14 

6i 

7 
8 

15     " 
i6     " 

7 
8 

i6     " 
17     " 

7 
8 

17  «      7 

18  «      8 

15     " 
i6     " 

9 

lO 

i6 
17 

9 

lO 

17     " 
i8     " 

9 

lO 

i8     " 
19     " 

9 

lO 

19     «      9 

,  20     "    10 

It  will  aid  tlie  pupil  in  learning  the  Subtraction  Table  to 
observe  that  it  is  the  reverse  of  Addition.  Tlius,  he  has  learned 
that  3  and  2  are  5.  Reversing  this,  he  will  see  that  2  from  5  leaves 
3,  etc.  Exercises  combining  the  two  tables  will  be  found  useful,  as  a 
review. 

CASE    I. 

35.  To  find  the  JDiffer€n€e,}N\\en  each  figure  of  the  Subtra- 
hend is  less  than  the  corresponding  figure  of  the  Minuend. 

Ux.  I.  From  568  dollars,  take  365  dollars. 

Analysis. — Let  the  numbers  be  set  down,  the  less 
tinder  the  greater,  as  in  the  margin.  Beginning  at 
the  right  hand,  we  proceed  thus :  5  units  from  8  units 
leave  3  units ;  set  the  3  in  units'  place  under  the 
figure  subtracted,  because  it  is  units.  Next,  6  tens 
from  6  tens  leave  o  tens.  Set  the  o  in  tens'  place 
under  the  figure  subtracted,  because  there  are  no  tens. 


OPJEKATION, 

$568    Min 

365 


Subt 


Rem. 


$203 
Finally,  1 


SUBTRACTION.  83 

hundreds  from  5  hundreds  leave  2  hundreds.  Set  the  2  in  hundreds' 
place,  because  it  is  hundreds.  The  remainder  is  $203.  All  similar 
examples  are  solved  in  like  manner. 

By  inspecting  this  illustration,  the  learner  will  discover 
the  following  principle : 

Units  of  the  same  order  are  subtracted  one  from 
another,  and  the  remainder  is  placed  under  the  figure  suo- 
tr acted.     (Arts.  9,  28.) 

Notes. — 1.  The  less  numher  is  placed  under  the  greater,  with 
units  under  units,  etc.,  for  the  sake  of  convenience  and  rapidity 
in  subtracting-. 

2.  Units  of  the  same  order  are  subtracted  from  each  other,  for 
the  same  reason  that  they  are  added  to  each  other.  (Art.  28,  n.) 

3.  We  subtract  the  figures  of  the  subtrahend  separately,  because  it 
is  easier  to  subtract  one  figure  at  a  time  than  several. 

4.  The  remainder  is  set  under  the  figure  subtracted,  because  it  is 
the  same  order  as  that  figure. 

(2.)         (3-)  (4.)  (50  (6.) 

From    648     726     8652     7230     9621 
Take    415    516    3440    4120    8510 


(7.)     (8.)     (9-)     (10.)    (II.) 

Prom        5304  6546  7852  8462  9991 

Take         II02  3214  5220  4130  8880 


(12.)  (13.)  (14.)  (15.) 

From        $6948  4315   ft-  I5346  9657  yds. 

Take        $2416  3212  ft.  $3106  5021  yds. 


(16.)  (17.)  (18.)  (19.) 

From       645921       8256072      72567803      965235804 
Take        435010      6135052      42103201      604132402 


20.  What  is  the  difference  between  3275  and  2132  ? 

21.  What  is  the  difference  between  5384  and  3264? 

22.  A  father  gave  his  sonl575o  and  his  daughter  ^4250; 
how  much  more  did  he  give  his  son  than  his  daughter? 


34  SUBTRACTIOif. 

CASE     II. 

36.  To  find  the  I>iffererice,  when  a  figure  in  the  Subtra- 
hend is  greater  than  the  corresponding  figure  of  the  Minuend, 

Ex.  I.  What  is  the  difference  between  8074  and  4869  ? 

ist  Analysis. — Let  tlie  numbers  be  set  down  as  in    ist  method. 
tlie  margin.      Beginning   at   the  right,   we  proceed     8074    liin. 
thus:    9   units  cannot   be  taken  from  4  units;   we     4860    Sub 

therefore  take   i  ten  from  the  7  tens  in  the  upper   

number,   and  add  it  to   4   units,  making   14.      Sub-    •720^    Rem, 
tracting  g  units  from  14  units  leaves  5  units,  which 
we  place  under  the  figure  subtracted.     Since  we  have  taken  i  ten 
from  7  tens,  there  are  but  6  tens  left,  and  6  tens  from  6  tens  leaves 

0  tens.  Next,  8  hundreds  cannot  be  taken  from  o  hundreds;  hence, 
we  take  i  thousand  from  the  8  thousand,  and  adding  it  to  the  o, 
we  have  10  hundreds.  Taking  8  hundreds  from  10  hundreds  leaves 
2  hundreds.  Finally,  4  thousand  from  7  thousand  (8 — i)  leave  3 
thousand.     The  remainder  is  3205. 

2d  Analysis. — Adding  10  to  4,  the  upper  figure,     2d  method. 
makes  14,  and  9  from  14  leaves  5.     Now,  to  balance      8074    Min. 
the   ten   added   to  the   upper   figure,   we  add    i    to     4860    Sub. 
the  next  higher  order  in  the  lower  number.     Adding     

1  to  6  makes  7,  and  7  from  7  leaves  o.  Next,  since  ^20=;  Rem 
we  cannot  take  8  from  o,  we  add  10  to  the  o,  and  8 

from  10  leaves  2.  Finally,  adding  i  to  4  makes  5,  and  5  from  8 
leaves  3.     The  remainder  is  3205,  the  same  as  before. 

Note. — The  chief  diflerence  between  the  two  methods  is  this :  In 
the  first,  we  subtract  i  from  the  next  figure  in  the  ujpper  number  ;  in 
the  second,  we  add  1  to  the  next  figure  in  the  lower  number.  The 
former  is  perhaps  the  more  philosophical ;  but  the  latter  is  mora 
convenient,  and  therefore  generally  practiced. 

By  inspecting  this  illustration,  it  will  be  seen, 

If  a  figure  in  the  lower  number  is  larger  than  that  above 

it,  we  add  10  to  the  upper  figure,  then  subtract,  and  add  i 

to  the  next  figure  in  the  lower  number. 

Rem  —Instead  of  adding  10  to  the  upper  figure,  many  prefer  to 

take  the  lower  figure  directly  from  10,  and  to  tlie  remainder  add  the 

upper  figure.     Thus,  9  from  10  leaves  i,  and  4  make  5,  etc. 

37.  Adding  10  to  the  upper  figure  is  called  borrowing  ; 
and  adding  i  to  the  next  figure  in  the  lower  number 
pays  it 


SUBTRACTION.  35 

I^oT£S. — I.  The  first  method  of  borrowing  depends  upon  the  obvi- 
ous principle  that  the  value  of  a  number  is  not  altered  by  transfer- 
ring  a  unit  from  a  higher  order  to  the  next  lower. 

2.  The  reason  that  the  second  method  of  horroicing  does  not  affect 
the  difference  between  the  two  numbers,  is  because  they  are  equally 
increased ,  and  when  two  numbers  are  equally  increased,  their  dif- 
ference is  not  altered. 

3.  The  reasor  for  borrowing  10,  instead  of  5,  8,  12,  or  any  other 
number,  is  because  ten  of  a  lower  order  are  equal  to  one  of  the  next 
higher.  If  numbers  increased  by  the  scale  of  5,  we  should  add  5  to 
the  upper  figure ;  if  by  the  scale  of  8,  we  should  add  8,  etc.    (Art.  g.) 

4.  We  begin  to  subtract  at  the  right,  because  when  we  borrow  we 
must  pay  before  subtracting  the  next  figure. 

(2-)         (3-)  (4-)  (5-)  (6.) 

From        784  842  6704  8042  9147 

Take         438  695  3598  5439  8638 

38.  The  preceding  principles  may  be  summed  up  in  the 
following 

GENERAL    RULE. 

I.  Place  the  less  number  under  tlie  greater,  units  under 
units,  etc. 

II.  Begin  at  the  right,  and  subtract  each  figure  in  tJia 
lower  number  from  the  one  above  it,  setting  the  remainder 
under  the  figure  subtracted.     (Art.  35.) 

III.  If  a  figure  in  the  lower  number  is  larger  than  the 
one  above  it,  add  10  to  the  upper  figure  ;  then  subtract,  and 
add  I  to  the  next  figure  in  the  loiver  number.     (Art.  s^.) 

39.  Peoof. — Add  the  remainder  to  the  subtrahend;  if 
the  sum  is  equal  to  the  minuend,  the  work  is  right. 

38.  How  write  numbers  for  subtraction?  How  proceed  when  a  figure  in  the 
lower  number  is  greater  tlian  the  one  above  it?  35.  Note.  Why  write  the  less 
number  under  the  greater,  etc.?  Why  subtract  the  figures  separately?  Why 
set  the  remainder  under  the  figure  subtracted  ?  37.  What  is  meant  by  borrow- 
ing ?  What  is  meant  by  paying  or  carrying  ?  Note.  Upon  what  principle  does 
the  first  method  of  borrowing  depend  ?  Why  does  not  the  second  method  of 
borrowing  aff"ect  the  diff'erence  between  the  two  numbers  ?  Why  borrow  10 
instead  of  5,  8,  12,  etc. ?  Why  begin  to  subtract  at  the  right  band?  ^9.  How 
prove  subtraction  ?    Note.  Upon  what  does  this  proof  depend  ? 


3G 


SUBTEACTIOK. 


Notes, — i.  This  proof  depends  upon  the  Axiom,  that  the  wJiole  u 
equal  to  the  sum  of  all  its  parts. 

2.  As  soon  as  the  class  understand  the  rule,  they  should  learn  to 
omit  all  words  but  the  results.  Thus,  .^n  Ex.  2,  instead  of  saying  9 
from  4  you  can't,  9  from  14  leaves  5,  etc.,  B&y  Jive,  naught,  three,  etc. 


EXAMPLES. 

Find  the  difference  between  84065  and  30428. 
(2.)  (3-)  (4.)  (5.) 

824  rods,        4523  pounds,        6841  years,        735^  acres, 
519  rods.        4456  pounds.        3062  years.       S^3^  acres. 


From 
Take 


(6.)- 

From  23941 
Take   1 2367 


(.0.) 

From  638024 
Take  403015 


(7.) 
46083 
23724 

(II.) 
7423614 
2414605 


(8.) 
52300 
25121 

(12.) 
8605240 
3062431 


(9.) 
483672 
216030 

(I3-) 

9042849 
6304120 


14.  From  85269  pounds,  take  33280  pounds. 

15.  From  412685  tons,  take  103068  tons. 

16.  From  840005  acres,  take  630651  acres. 

17.  What  is  the  difference  between  95301  and  60358? 

18.  What  is  the  difference  between  1675236  and  439243  ? 

19.  Subtract  2036573  from  5670378. 

20.  Subtract  35000384  from  68230460. 

21.  Subtract  250600325  from  600230021. 

22.  A  man  bought  a  house  and  lot  for  $36250,  and 
paid  $17260  down :  how  much  does  he  still  owe  ? 

23.  A  man  bought  a  farm  for  I19200,  and  sold  it  for 
$17285  :  what  was  his  loss? 

24.  A  merchant  bought  a  cargo  of  tea  for  1 1265 235,  and 
Bold  it  for  I1680261 :  what  was  his  gain? 

25.  A's  income  is  $645275,  and  B's  $845280:  what  is 
the  difference  in  their  incomes  ? 


SUBTRACTIOK^.  3? 

26.  A   bankrupt's   assets  are   $569257,  and  his   debts 
$849236  :  how  much  will  his  creditors  lose  ? 

27.  Subtract  9999999  from  iiiniio. 

28.  Subtract  666666666  from  7000000000. 

29.  Subtract  8888888888  from  loooooooooo. 

30.  Take  200350043010  from  490103060756. 

31.  Take  53100060573604  from  80130645002120. 

32.  Take  675000000364906  from  901638000241036. 
;^;^,  From  two  millions  and  five,  take  ten  thousand. 

34.  From  one  million,  take  forty-five  hundred. 

35.  From  sixty-five  thousand  and  sixty-five,  take  five 
hundred  and  one. 

36.  From  one  billion  and  one  thousand,  take  one  million. 

37.  Our  national  independence  was  declared  in  1776: 
how  many  years  since  ? 

38.  Our  forefathers  landed  at  Plymouth  in  1620:  how 
many  years  since  ? 

39.  A  father  having  3265  acres  of  land,  gave  his  son 
1030  acres :  how  many  acres  has  he  left  ? 

40.  The  Earth  is  95300000*  miles  from  the  sun,  and 
Venus  68770000  miles:  required  the  difierence. 

41.  Washington  died  in  1799,  at  the  age  of  67  years:  in 
what  year  was  he  born  ? 

42.  Dr.  Franklin  was  born  in  1706,  and  died  in  1790: 
at  what  ago  did  he  die  ? 

43.  Sir  Isaac  Newton  died  in  1727,  at  the  age  of  85 
years :  in  what  year  was  he  born  ? 

44.  Jupiter  is  496000000  miles  from  the  sun,  and 
Saturn  909000000  miles :  what  is  the  difierence  ? 

45.  In  1865,  the  sales  of  A.  T.  Stewart  &  Co.,  by  ofiicial 
returns,  were  $39391688;  those  of  H.  B.  Claflin  &  Co., 
$42506715  :  what  was  the  difference  in  their  sales? 

46.  The  population  of  the  United  States  in  i860  was 
31445080;  in  1850  it  was  23 191 87 6:  what  was  the  increase  ? 

*  Kiddle's  Astronomy. 


SUBTRACTlOlf.  38 


QUESTIONS    FOR     REVIEW. 

1.  A  man's  gross  income  was  Si 5 65  a  month  for  two 
successive  months,  and  his  outgoes  for  the  same  time 
8965  :  what  was  his  net  profit  ? 

2.  A  man  paid  I2635  for  his  farm,  and  $758  for  stock; 
he  then  sold  them  for  $4500 :  what  was  his  gain  ? 

3.  A  flour  dealer  has  3560  barrels  of  flour;  after  selling 
1380  barrels  to  one  customer,  and  985  to  another,  how 
many  barrels  will  he  have  left  ? 

4.  If  I  deposit  in  bank  $6530,  and  give  three  checks  of 
$733  each,  how  much  shall  I  have  left  ? 

5.  What  is  the  difference  between  3658  +  256  +  4236  and 
2430  +  1249? 

6.  What  is  the  difference  between  6035+560  +  75  and 
5003  +  360  +  28? 

7.  What  is  the  difference  between  891+306  +  5007  and 
40  +  601  +  1703  +  89? 

8.  What  is  the  difference  between  900130  +  23040  and 
19004+  100607  ? 

9.  A  man  having  I16250,  gained  $3245  by  specula- 
tion, and  spent  I5203  in  traveling:  how  much  had  he 
left? 

10.  A  farmer  bought  3  horses,  for  which  he  gave  $275, 
$320,  and  $418  respectively,  and  paid  $50  down:  how 
much  does  he  still  owe  for  them  ? 

11.  A  young  man  received  three  legacies  of  $3263, 
I5490,  and  $7205  respectively;  he  lost  I4795  minus  I1360 
by  gambling :  how  much  was  he  then  worth  ? 

12.  What  is  the  difference  between  6286  +  850  and 
6286-850? 

13.  What  is  the  difference  between  11 325  — 2361  and 
8030  —  3500? 

14.  What  number  added  to  103256  will  make  215378? 

15.  What  number  added  to  573020  will  make  700700  ? 


SUBTEACTION".  39 

i6.  What  number  subtracted  from  230375  will  leaye 
121487  ? 

17.  What  number  subtracted  from  317250  will  leaye 
190300  ? 

18.  If  the  greater  of  two  numbers  is  59253,  and  the 
difference  is  21 231,  what  is  the  less  number? 

19.  If  the  difference  between  two  numbers  is  1363,  and 
the  greater  is  45261,  what  is  the  less  number? 

20.  The  sum  of  two  numbers  is  63270,  and  one  of  them 
is  29385  :  what  is  the  other? 

21.  What  number  increased  by  2343  —  131,  will  become 
3265-291  ? 

22.  What  number  increased  by  3520+  1060,  will  become 

6539-279? 

23.  What  number  subtracted  from  5009,  will  become 

23404-471? 

24.  Agreed  to  pay  a  carpenter  $5260  for  building  a 
house;  $3520  for  the  masonry,  and  $1950  for  painting: 
how  much  shall  I  owe  him  after  paying  $6000  ? 

25.  If  a  man  earns  $150  a  month,  and  it  costs  him  $6;^ 
a  month  to  support  his  family,  how  much  will  he 
accumulate  in  6  months  ? 

26.  A's  income-tax  is  $1165,  B's  is  $163  less  than  A's; 
and  C's  is  as  much  as  A  and  B^s  together,  minus  $365 : 
what  is  C's  tax  ? 

27.  A  is  worth  $15230,  B  is  worth  $1260  less  than  A; 
and  0  is  worth  as  mucli  as  both,  wanting  $1760:  what  is 
B  worth,  and  what  C  ? 

28.  The  sum  of  4  numbers  is  45260;  the  first  is  21000, 
the  second  8200  less  than  the  first,  the  third  7013  less 
than  the  second  :  what  is  the  fourth  ? 

29.  A  sailor  boastingly  said ;  If  I  could  save  I263 
more,  I  should  have  ^1000:  how  much  had  he? 

30.  The  difference  between  A  and  B's  estates  is  $1525  ; 
B,  who  has  the  least,  is  worth  $17250:  what  is  A  worth  ? 


MULTIPLICATION. 

40.  Multiplication  is  finding  the  amount  of  a 

number  taken  or  added  to  itself,  a  given  number  of  times. 

Tlie  Multiplicand  is  tlie  number  to  be  multiplied. 

The  Multiplier  is  the  number  by  which  we  mul- 
tiply. 

The  Product  is  the  number  found  by  multiplici*tion. 
Thus,  when  it  is  said  that  3  times  6  are  18,  6  is  the  mul- 
tiplicand, 3  the  multiplier,  and  18  the  product. 

41.  The  multiplier  shows  liow  many  times  the  multi- 
plicand is  to  be  taken.     Thus, 

Multiplying  by  i  is  taking  the  multiplicand  once  ; 
Multiplying  by  2  is  taking  the  multiplicand  twice  ;  and 
Multiplying  by  any  whole  number  is  taking  the  multi- 
plicand as  many  times  as  there  are  units  in  the  multiplier. 

Notes. — i.  The  term  wvltiplication,  from  the  Latin  multipUco, 
multus,  many,  and  plico,  to  fold,  primarily  signifies  to  increase  by 
regular  accessions. 

2.  Multiplicand,  from  the  same  root,  signifies  that  which  is  to  he  tnul- 
tiphed  ;  the  termination  nd,  having  the  force  of  to  he.    (Art.  33,  n.) 

42.  The  multiplier  and  multiplicand  are  also  called 
Factors;  because  they  make  or  produce  the  product 
Thus,  2  and  7  are  the  factors  of  14. 

Notes. — i.  The  term/ac^or,  is  from  the  Latin /aa^?,  to  produce. 
2.  When  the  multiplicand  contains  only  one  denomination,  the 
operation  is  called  Simple  Multiplication. 

40.  What  ie  multiplication?  What  1e  the  numher  multiplied  called?  The 
mimher  to  multiply  hy  ?  The  result  ?  When  it  is  said,  3  times  4  are  12,  which  is 
the  multiplicand?  The  multiplier?  The  product?  41.  What  does  the  multi- 
plier show  ?  What  is  it  to  multiply  by  i  ?  By  2  ?  42.  What  else  are  the  nnmhers 
to  be  multiplied  together  called  ?  WTiy  ?  NoU.  Meaninsr  of  the  term  factor  ? 
What  is  the  operation  called  when  the  multiplicand  contains  only  one  denomi- 
nation ?  43.  The  sign  of  multiplication. 


MULTIPLICATION, 


41 


43.  The  Sign  of  Multiplication  is  an  oblique 
cross  (  X  ),  placed  between  the  factors.  Thus,  7x5  means 
that  7  and  5  are  to  be  multiplied  together,  and  is  read 
"  7  times  5,"  "  7  multiplied  by  5,"  or  "  7  into  5." 

44.  Numbers  subject  to  the  same  operation  are  placed 
within  ?i  parenthesis  ( ),  or  under  a  line  called  a  vinculum 

( ).     Thus  (4  +  5)  X  3,  or  4  +  5  X  3,  shows  that  the  sum 

of  4  and  5  is  to  be  multiplied  by  3. 

MULTIPLICATION  TABLE. 


2  times 

1  are    2 

2  "     4 

3  "     6 

4  «     8 

5 
6 

7 
8 

9 

10 
II 
12 


3  times 

1  are    3 

2  "  6 

3  "  9 

4  "  12 

5  "  15 

6  "  18 

7  "  21 

8  "  24 

9  "  27 

0  "  30 

1  "  33 

2  "  7,6 


times 

are 

4 

a 

8 

a 

12 

a 

16 

a 

20 

a 

24 

a 

28 

(i 

32 

a 

36 

a 

40 

a 

44 

a 

48 

5  times 

1  are   5 

2  "   10 

a 


15 

20 

25 
30 

35 
40 

45 
50 
55 
60 


6  times 
I  are   6 


2 
3 

4 

5 
6 

7 
8 

9 
10 
II 
12 


12 
18 
24 

30 
36 
42 
48 

54 
60  I 
66;i; 

72  \i: 


7  times 

1  are    7 

2  "   14 


21 

28 

35 
42 
49 

56 

^Z 

70 

77 
^4 


8  times 

9  times 

10  times 

II  times 

12  times 

I  are 

8 

I  are 

Q 

I  are  10 

I  are   11 

I  are  12 

2    " 

16 

2    " 

18 

2    ' 

20 

2    "     22 

2    "     24 

3    " 

24 

3    " 

27 

3    ' 

30 

3    "     Z2> 

3    "     36 

4    " 

32 

4    " 

36 

4   ' 

40 

4    "     44 

4    "     48 

5    " 

40 

5    " 

45 

5    ^ 

50 

5    "     55 

5    "     60 

6    " 

48 

6    " 

54 

6    ^ 

60 

6    «     66 

6    -     72 

7    " 

S^ 

7    " 

63 

7    ^ 

70 

7    "     77 

7    "     84 

8    " 

64 

8    " 

72 

8    ' 

80 

8    "     88 

8    "     96 

9    " 

72 

9    " 

81 

9    ^ 

90 

9    "     99 

9    "  108 

10    " 

80 

10    " 

90 

10    ' 

100 

10    "   no 

10    "  120 

II     " 

88 

II    « 

90 

II    ' 

1 10 

II    "121 

II    "  132 

12    « 

96 

12    " 

108 

12    ' 

120 

12    "  132 

12    «  144 

Tlie  pupil  will  observe  tliat  the  'products  by  5,  terminate  iq 
5  and  o,  alternately.     Tbus,  5  times  i  are  5 ;  5  times  2  arc  10. 


0 

^ 

^ 

Si 

m 

^ 

^ 

» 

m 

i5 

0 

m 

4:2  MULTIPLICATION. 

The  products  by  lo  consist  of  the  figure  multiplied  and  a  cipher.- 
Thus,  10  times  i  are  lo ;  ten  times  2  are  20,  and  so  on. 

The  first  nine  products  by  11  are  formed  by  repeating  the  figure 
multiplied.     Thus,  11  times  i  are  11 ;  11  times  2  are  22,  and  so  on. 

The  first  figure  of  the  first  nine  products  by  9  increases,  and  the 
second  decreases  regularly  by  i ;  while  the  sum  of  the  digits  in  each 
product  is  9.     Thus,  9  times  2  are  18,  9  times  3  are  27,  and  so  on. 

45.  When  the  factors  are  abstract  numbers, 
it  is  immaterial  in  what  order  they  are  multi- 
plied. Thus,  the  product  of  3  times  4  is  equal 
to  4  times  3.     For,  taking.  4  units  3  times,  is 

the  same  as  taking  3  units  4  times ;  that  is,  4  x  3  =  3  x  4. 

Notes. — i.  It  is  more  convenient  and  therefore  customary,  to 
take  the  larger  of  the  two  given  numbers  for  the  multiplicand. 
Thus,  it  is  better  to  multiply  5276  by  8,  than  8  by  5276. 

2.  Multiplication  is  the  same  in  principle  as  Addition;  and  is 
sometimes  said  to  be  «-  short  method  of  adding  a  number  to  itself  a 
given  number  of  times.  Thus:  4  stars +  4  stars +  4  stars=3  times  4 
or  12  stars. 

46.  The  iwoduct  is  the  same  name  or  hind  as  the  multi- 
plicand. For,  repealing  a  number  does  not  change  its 
nature.   Thus,  if  we  repeat  dollars,  they  are  still  dollars,  etc. 

47.  The  multiplier  must  be  an  abstract  number,  or  con- 
sidered as  such  for  the  time  being.  For,  the  multiplier 
shows  Jiow  many  times  the  multiplicand  is  to  be  taken ;  and 
to  say  that  one  quantity  is  taken  as  many  times  as  another 
\s  heavy — is  ait  urd. 

Note. — When  it  is  asked  what  25  cts.  multiplied  by  25  cts.,  or 
2S.  6d.  by  2s.  6d ,  will  produce,  the  questions,  if  taken  literally, 
are  nonsense.  For,  2s.  6d.  cannot  be  repeated  2s.  6d.  times,  nor  25  cts. 
25  cts.  times ;  but  we  can  multiply  25  cts.  by  the  number  25,  and 
2S.  6d.  by  2\.  In  like  manner  we  can  multiply  the  price  of  i  yard  by 
the  number  of  yards  in  an  article,  and  the  product  will  be  the  cost. 

45.  When  the  factors  are  abstract,  docs  it  make  any  difference  with  the  pro- 
duct which  is  taken  for  the  multiplicand  ?  Note.  Which  is  commonly  taken  f 
Why  ?  What  is  Multiplication  the  same  as  ?  46.  What  denomination  is  the  pra 
duct  ?    Why  ?    47.  "What  must  the  multiplier  be  ?    Why  ? 


MULTIPLICATION.  43 

CASE    I. 

48.  To  find   the  Product  of  two    numbers,  when   the 

Multiplier  has  but  one  figure. 

I.  What  is  the  cost  of  3  horses,  at  $286  apiece  ? 
Analysis. — 3  horses  will  cost  3  times  as  mucli  as         opebation. 

1  horse.  Let  the  numbers  be  set  down  as  in  the  $286  Multd. 
margin.     Beginning  at  the  right,  we  proceed  thus :  ^  Mult. 

3  times  6  units  are  18  units.     We  set  the  8  in  units'        

place  because  it  is  units,  and  carry  the  i  ten  to  the  $858  Prod, 
product  of  the  tens,  as  in  Addition. (Art.  29.)    Next, 

3  times  8  tens  are  24  tens  and  i  (to  carry)  make  25  tens,  or  2  hun- 
dred  and  5  tens.  We  set  the  5  in  tens'  place  because  it  is  tens,  and 
carry  the  2  hundred  to  the  product  of  hmidreds.     Finally,  3  times 

2  hundred  arc  6  hundred,  and  2  (to  carry)  are  8  himdred.  We  sot 
the  8  in  hundreds'  place  because  it  is  hundreds.  Therefore  the  3 
horses  cost  $858.     All  similar  examples  are  solved  in  like  manner. 

By  inspecting  the  preceding  analysis,  the  learner  will 
discover  the  following  principle : 

Each  figure  of  the  multiplicand  is  multiplied  hj  the 
multiplier,  heginning  at  the  rigM,  and  the  result  is  set 
do2vn  as  in  Addition.     (Art.  29.) 

Notes. — i.  The  reason  for  setting  the  multiplier  under  the  multi- 
plicand is  simply  for  convenience  in  muliiplyiug. 

2.  The  reasons  for-  beginning  to  multiply  at  the  right  hand,  as 
well  as  for  setting  down  the  units  and  carrying  the  tens,  are  tho 
same  as  those  in  Addition.     (Art.  29,  n.) 

3.  In  reciting  the  following  examples,  the  pupil  should  care- 
fully analyze  each  ;  then  give  the  results  of  the  operations  re- 
quired. 

49.  Units  multiplied  hy  miits,  it  should  be  observed, 
produce  imits  ;  tens  by  units,  produce  tens ;  hundreds  by 
units,  produce  hundreds  ;  and,  universally, 

If  the  multiplying  figure  is  units,  the  product  will  be 
the  same  order  as  the  figure  multiplied. 
Again,  if  the  figure  multiplied  is  units,  the  product  will 

40.  What  do  units  multiplied  by  units  produce  ?  Tens  by  units  ?  Hundreds 
V  units  ?    When  the  multiplying  fl^re  is  units,  what  is  the  product  J 


44  MULTIPLICATION^, 

be  the  same  order  as  the  multiplying  figure;  for  the 
product  is  the  same,  whichever  factor  is  taken  for  the 
multiplier.     (Art.  45.) 

(^•)  (3-)  (4-)  (5-) 

Multiply    2563  13278  2648203  48033265 

By  4  5  6  7 


6.  What  will  8  carriages  cost,  at  $750  apiece? 

7.  What  cost  9  village  lots,  at  $1375  a  lot? 

8.  At  $3500  apiece,  what  will  10  houses  cost? 

9.  At  $865   a   hundred,  how  much  will    11  hundred- 
weight of  opium  come  to  ? 

10.  If  a  steamship  sail  358  miles  per  day,  how  far  will 
she  sail  in  1 2  days  ? 

11.  Multiply  86504  by  5.        12.  Multiply  803127  by  7. 
13.  Multiply  440210  by  6.      14.  Multiply  920032  by  8. 
15.  Multiply  603050  by  9.      16.  Multiply  810305  by  10. 
17.  Multiply  753825  by  II.     18.  Multiply  954635  by  12. 

19.  What  cost  4236  barrels  of  apples,  at  $7  a  barrel  ? 

20.  What  cost  5167  melons,  at  11  cents  apiece? 

21.  At  6  shillings  apiece,  what  will  1595  arithmetics 
cost? 

22.  At  $12  a  barrel,  what  will  be  the  cost  of  1350  bar- 
rels of  flour  ? 

23.  Wliat  will  1735  boxes  of  soap  cost,  at  $9  a  box  ? 

24.  When  peaches  are  1 2  shillings  a  basket,  what  will 
2363  baskets  cost  ? 

25.  If  a  man  travels  8  miles  an  hour,  how  far  will  he 
travel  in  3260  hours  ? 

26.  A  builder  sold   10  houses  for  $4560  apiece:   how 
much  did  he  receive  for  them  all  ? 

27.  What  will  it  cost  to  construct  11  miles  of  railroad, 
at  $12250  per  mile? 

28.  What  will  be  the  expense  of  building  12  ferry-boats, 
at  I23250  apiece  ? 


MULTIPLICATIOK.  45 

CASE    II. 

50.     To    find    the    Product    of    two    numbers,    when    the 
Multiplier  has  two  or  more  figures. 

I.  A  manufacturer  bought  204  bales  of  cotton,  at  $176 
a  bale:  what  did  he  pay  for  the  cotton  ? 

Analysis. — Write  the  numbers  as  in  the  mar-        $176  Multd. 
gin.     Beginning  at  the  right :  4  times  6  units  are  204  Mult 

24  units,  or  2  tens  and  4  units.     Set  the  4  in  units'         

place,  and  carry  the  2  to  the  next  product.     4  7^4 

times  7  tens  are  28  tens  and  2  are  30  tens,  or  3      3^2 

hundred  and  o  tens.     Set  the  o  in  tens'  place  and  ■ 

carry  the  3  to  the  next  product.  4  times  i  hun-  $35904  Ans. 
are  4  hun.  and  3  are  7  hun.  Set  the  7  in  hundreds' 
place .  The  product  by  o  tens  is  o ;  we  therefore  omit  it.  Again, 
2  hundred  times  6  units  are  12  hun.  units,  equal  to  i  thousand  and 
2  hundred.  Set  the  2  in  hundreds'  place  and  carry  the  i  to  the 
next  product.  2  hundred  times  7  tens  are  14  hundreds  of  tens,  equal 
to  14  thousand,  and  i  are  15  thousand.  Set  the  5  in  thousands' 
place,  and  carry  the  i  to  the  next  product,  and  so  on.  Adding 
these  results,  the  sum  is  the  cost. 

Remakk. — The  results  which  arise  from  multiplying  the  multipli- 
cand by  the  separate  figures  of  the  multiplier,  are  called  partial  pro- 
ducts ;  because  they  are  parts  of  the  whole  product. 

By  inspecting  this  analysis,  the  learner  will  discover, 

1.  Tlie  multiplicand  is  multiplied  hy  eacli  figure  of  the 
multiplier,  hegin^iing  at  the  right,  and  the  result  set  down 
as  in  Addition. 

2.  The  first  figure  of  each  partial  product  is  placed 
under  the  multiplying  figure,  and  the  sum  of  the  partial 
products  is  the  true  answer. 

Notes. — i.  When  there  are  ciphers  between  the  significant 
figures  of  the  multiplier,  omit  them,  and  multiply  by  the  next  sig- 
nificant figure. 

2.  We  multiply  by  each  figure  of  the  multiplier  separately,  when  it 
exceeds  12,  for  the  obvious  reason,  that  it  is  not  convenient  to  multi- 
ply by  the  whole  of  a  large  number  at  once. 

3.  The  first  figure  of  each  partial  product  is  placed  under  the 
multiplying  figure;  because  it  is  the  same  order  as  that  figure. 

4.  The  several  partial  prof\ucts  are  added  together,  because  the 
whole  product  is  equal  to  the  sam  of  aU  its  parts. 


46  MULTIPLICATION. 


(^■) 

(3.) 

(4.) 

(5.) 

Multiply    426 

5^3 

1248 

2506 

By                 24 

35 

52 

304 

51.  The  preceding  principles  may  be  summed  up  in 
the  following 

GENERAL    RULE 

I.  Place  the  muUijMer  under  tlie  multiplicand,  units 
under  units,  etc. 

II.  When  the  multiplier  has  hut  one  figure,  heginning  at 
the  right,  multiply  each  figure  of  the  multiiMca^id  iy  it, 
and  set  down  the  result  as  in  Addition. 

III.  If  the  multiplier  has  two  or  more  figures,  multiply 
the  multiplicand  hy  each  figure  of  the  multiplier  separately, 
and  set  the  first  figure  of  each  partial  product  under  the 
multiilying  figure. 

Finally,  the  sum  of  the  partial  products  will  he  the 
ansiuer  required. 

Note. — The  pupil  should  early  learn  to  abbreviate  the  several 
steps  in  multiplying  as  in  Addition.     (Art.  31,  7i.) 

PROOF. 

52.  By  Multiplication. — Multiply  the  multiplier  hy  the 
multiplicand  ;  if  this  result  agrees  ivith  the  first,  the  ivork 
is  right. 

Note. — This  proof  is  based  upon  the  principle  that  the  result 
will  be  the  same,  whichever  number  is  taken  as  the  multiplicand. 

53.  By  excess  of  9s. — Find  the  excess  of  gs  in  each  factor 
separately ;  then  multiply  these  excesses  together,  and  reject 
the  gs  from  the  result ;  if  this  excess  agrees  with  the  excess 
jf  9s  in  the  answer,  the  work  is  right. 

51.  How  write  numbers  for  multiplication  ?  When  the  multiplier  has  hut  one 
figure,  how  proceed  ?  When  it  haa  two  or  more  ?  Wliat  ie  finally  done  with  the 
partial  products  ?  4S.  Note.  Why  write  the  multiplier  under  the  multiplicand, 
units  under  units  ?  Why  begin  at  the  right  hand  ?  50.  W^hat  are  partial  products  ? 
Note.  Why  multiply  by  each  figure  separately  ?  Why  set  the  first  figure  of  each 
Tinder  the  multiplying  figure  ?  Why  add  the  several  partial  products  together  ? 
52.  How  prove  multiplicetion  by  multiplication  ?  Note.  Upon  what  ie  this  proof 
based  ?    53.  How  prove  it  by  excess  of  98  ? 


MULT  IPLICATIOS-, 


47 


Note. — This  method  of  proof,  if  deemed  advisable,  may  be  omitted 
till  review.  It  is  placed  here  for  convenience  of  reference.  Though 
depending  on  a  peculiar  property  of  numbers,  it  is  easily  applied, 
and  is  confessedly  the  most  expediitous  method  of  proving  multipli- 
cation yet  devised. 

54.  To  find  the  Excess  of  98  in  a  number. 

Beginning  at  the  left  hand,  add  the  figures  together,  and  as  soon 
as  the  sum  is  9  or  more,  reject  9  and  add  the  remainder  to  the  next 
figure,  and  so  on. 

Let  it  be  required  to  find  the  excess  of  9s  in  7548467. 

Adding  7  to  5,  the  smn  is  12.  Rejecting  9  from  12,  leaves  3  ;  and 
3  added  to  4  are  7,  and  8  are  15.  Rejecting  9  from  15,  leaves  6 ;  and 
6  added  to  4  are  10.  Rejecting  9  from  10,  leaves  i ;  and  i  added  to 
6  are  7,  and  7  are  14.  Finally,  rejecting  9  from  14  leaves  5,  the 
excess  required. 

Notes. — i.  It  will  be  observed  that  the  excess  of  9s  in  any  two 
digits  is  always  equal  to  the  sum,  or  the  excess  in  the  sum,  of  those 
digits.  Thus,  in  15  the  excess  is  6,  and  1  +  5=6;  so  in  51  it  is  6, 
and  5  +  1=6.     In  56  the  sum  is  11,  the  excess  2. 

2.  The  operation  of  finding  the  excess  of  9s  in  a  number  is  called 
casting  out  the  gs. 

EXAMPLES. 

I.  Wliat  is  the  product  of  746  multiplied  by  475  ? 


OPKBATION. 
746 

475 


3730 
5222 
2984 


Proof  by  Excess  of  gs. 

Excess  of  9s  in  multd.  is  8 
"  9  s  in  mult.    "  7 

Now 8  X  7  =  56 

The  excess  of  9s  in  56  is  2 
The  excess  of  9s  in  prod,  is  2 
And 2  =  2 


Proof  by  Mult. 

475 
746 


2850 
1900 
3325 


Ans. 

'  354350 

■^ns.  354350 

(2.) 

(3.) 

(4.) 

(5.) 

Multiply 

5645 

18934 

48367 

231456 

By 

43 

65 

75 

87 

54.  How  And  the  excess  of  98  ? 


48  MULTIPLICATION. 


(6.) 

(7.) 

(8.) 

(9.) 

Multiply 

1421673 

2342678 

4392460 

5230648 

By 

234 

402 

347 

526 

10.  Mult.  640231  by  205.  II.  Mult.  520608  by  675. 

12.  Mult.  431220  by  1234.  13.  Mult.  623075  by  2650. 

14.  Mult.  730650  by  2167.  15.  Mult.  593287  by  6007. 

r6.  Mult.  843700  by  3465,  17.  Mult.  748643  by  2100. 

18.  Mult.  9000401  by  50001.  19.  Mult.  82030405  by  23456. 

20.  How  many  pounds  m  1375  chests  of  tea,  each  chest 
containing  6;^  pounds  ? 

21.  What  cost  738  carts,  at  I75  apiece? 

22.  At  43  bushels  per  acre,  how  many  bushels  of  wheat 
will  520  acres  produce? 

23.  At  $163  apiece,  what  will  be  the  cost  of  1368  covered 
buggies  ? 

24.  If  a  man  travel  215  miles  per  day,  how  far  can  he 
travel  in  365  days  ? 

25.  There  are  5280  feet  in  a  mile:  how  many  feet  in 
256  miles? 

26.  What  cost  21 15  revolvers,  at  I23  apiece? 

27.  Bought  1978  barrels  of  pickles,  at  $17  ;  how  much 
did  they  come  to  ? 

28.  If  railroad  cars  are  $4735  apiece,  what  will  be  the 
expense  of  500  ? 

29.  How  far  will  2163  spools  of  thread  extend,  each 
containing  25  yards? 

30.  Bought   15265  ambulances,  at  $117  apiece:  what 
was  the  amount  of  the  bill  ? 

31.  What  will  3563  tons  of  railroad  iron  cost,  at  168 
per  ton  ? 

32.  How  far  will  a  man  skate  in  6  days,  allowing  he 
skates  8  hours  a  day,  and  goes  y  miles  an  hour? 


MULTIPLICATION^.  4^ 

CONTRACTIONS. 

55.  A  Composite  NtiTnher  is  the  product  of  two 
or  more  factors,  each  of  which  is  greater  than  i.  Thus 
15=3  X  5,  and  42=^2  x  3  x  7,  are  composite  numbers. 

Notes. — i.  The  product  of  a  number  multiplied  into  itself  is 
called  a  power.    Thus,  9=3  x  3,  is  a  power. 

2.  Composite  is  from  tlie  Latin  compono,  to  place  together. 

3.  The  term^  factors  oxi^  parts  must  not  be  confounded  with  each 
other.  The  former  are  multiplied  together  to  produce  a  number ; 
r'lie  latter  are  added.  Thus,  3  and  5  are  tlxe  factors  of  15  ;  but  5  and 
10,  6  and  9,  7  and  8,  etc.,  are  the  parts  of  15. 

1.  What  are  the  factors  of  45  ?     Ans.  s  and  9. 

2.  What  are  the  factors  of  24  ?    Of  27?    Of  28  ?  Of  30? 

3.  What  are  the  factors  of  32?    Of  35  ?    Of  42?  Of48.f' 

4.  What  are  the  factors  of  5 4  ?    Of  63  ?    Of  7  2  ?  Of  84  ? 


CASE    I.    ^ 
56.  To  multiply  by  a  Cofnposite  Number. 

I.  A  farmer  sold  15  boxes  of  butter,  each  weighing  20 
pounds :  how  many  pounds  did  he  sell  ? 

Analysis. — 15  =  5  times  3;  hence,  15  boxes 
will  weigh  5  times  as  much  as  3  boxes.  Now, 
if  I  box  weighs  20  pounds,  3  boxes  will  weigh 


OPEEATIOIT. 

20  pounds. 


3  times  20,  or  60  pounds.     Agam,  if  3  boxes  ^ 


60 


weigh  60  pounds,  5  times  3  boxes  will  weigh  5 
times  60,  or  300  pounds.  He  therefore  sold  300 
T>ounds.     In  the  operation,  we  first  multiply  by.  5 

the  factor  3,  and  the  product  thus  arising  by  the  

-other  factor  5.     Hence,  the  Ans.  300  pounds, 

EuLE. — Multiply  the  multiplicand  by  one  of  the  factors 
of  the  multiplier,  then  this  product  hy  another,  and  so  on, 
till  all  the  factors  have  been  used. 

The  last  product  will  he  the  answer, 

35.  A  composite  number?  Note.  The  difference  between  fiactors  ano  parts! 
;6  Hew  multiply  by  a  compoaiie  number  ? 

z 


50  MULTIPLICATION. 

Notes. — i.  This  rule  is  based  upon  the  principle  that  it  is 
immaterial  in  what  order  two  factors  are  multiplied.     (Art.  45.) 

The  samt,  illustration  may  be  extended  to  three  or  more  numbers. 
For,  the  product  of  two  of  the  factors  may  be  considered  as  one 
Qumber,  and  this  may  be  used  before  or  after  a  third  factor,  etc. 

2.  The  process  of  multiplying  three  or  more  factors  together,  is 
called  Continued  Multiplication ;  the  result,  the  continued  product. 

2.  Multiply  $568  by  35.    Ans.  $19880. 

3.  Multiply  2604  by  25.        4.  Multiply  6052  by  48. 
5.  Multiply  8091  by  63.         6.  Multiply  45321  by  72. 

7.  In  I  cubic  foot  there  are  1728  cubic  inches:  how 
many  cubic  inches  are  there  in  84  cubic  feet  ? 

8.  If  a  ton  of  copper  ore  is  worth  $5268,  what  is  the 
worth  of  56  tons? 

9.  What  cost  125  houses,  at  $1580  apiece? 


CASE    II. 

57.  To  multiply  by  lO,  100,  lOOO,  etc. 

10.  What  will  100  cows  cost,  at  $31  apiece? 

Analysis. — Annexing  a  cipher  to  a  number  moves  each  figure  one 
place  to  the  left ;  but  moving  a  figure  one  place  to  the  left  increases 
ioS  value  ten  times ;  therefore  annexing  a  cipher  to  a  number 
multiplies  it  by  10.  In  like  manner,  annexing  two  ciphers,  multi- 
plies it  by  a  hundred,  etc.  (Art,  12.)  Therefore,  $31  x  ioo=$3ioa 
Hence,  the 

Rule. — Annex  as  mamj  ciphers  to  the  multiplicand  as 
there  are  ciphers  i?i  the  multijMer. 

Note. — The  term  annex,  -from  the  Latin  ad  and  ne^to,  to  join  tc^ 
eignifies  to  place  after. 

11.  What  cost  1000  horses,  at  I356  apiece? 

12.  Multiply  40530  by  1000. 

13.  Multiply  9850685  by  loooo. 

14.  Multiply  84050071  by  1 00000.  * 

15.  Multiply  360753429  by  loooooo. 

.57.  How  maltiply  by  to,  too,  1000,  etc.  T 


MULTIPLICATION.  51 


CASE    III. 


58.  To  multiply,  when  there  are  Ciphers  on  the  right  of 
either  or  both  Factors. 

20.  What  is  the  product  of  87000  multiplied  by  230  ? 

Analysis. — The  factors  of  the  ruuhiplicand  are  87000 

87  and  1000;  the  factors  of  the  multiplier  are  23  2^0 

and  10.     We  first  multiply  the  factoi*s  consisting  — r 

of  significant  figures  ;  then  multiply  this  product  ^^ 

by  the  other  two   factors   (1000 x  10),  or  loooo,  ^74 

by  annexing  4  ciphers  to  it.     Hence,  the  -4 ?^5.  20010000 

EuLE. — Multiply  the  significant  figures  together ;  and 
to  the  result  annex  as  many  ciphers  as  are  found  on  the 
right  of  doth  factors. 

Note. — This  rule  is  based  upon  the  two  preceding  cases;  for,  by 
supposition,  one  or  both  the  given  numbers  are  composite ;  and  one 
of  the  factors  of  this  composite  number  is  10,  100,  etc. 


(21.) 

(22.) 

(23.) 

Multiply  2130 

64000 

83046 

By        700 

52 

2000 

24.  In  I  barrel  of  pork  there  are  200  pounds:  how  many 
pounds  in  3700  barrels? 

25.  What  will  2300  liead  of  cattle  cost,  at  $80  per  head? 

26.  In  $1  there  are  100  cts. :  how  many  cts.  in  I26000? 

27.  The  salary  of  the  President  is  I50000  a  year:  how 
much  will  it  amount  to  in  21  years  ? 

28.  If  a  clock  ticks  86400   times  in  i  day,  how  many 
times  will  it  tick  in  7000  days? 

29.  If  one  ship  costs  $150000,  what  will  49  cost? 

30.  670103700x60030040? 

31.  800021000x80002100? 

32.  570305000x40000620? 


58.  When  one  or  both  factors  have  ciphers  on  the  right  ?    Note.  Upon  what  is 
this  rule  based  ? 


53  MULTIPLICATION. 

^^.  467234630X27000000? 

34.  890000000x350741237? 

35.  9400000027  X  28000000  ? 

36.  Multiply  39  millions  and  200  thousand  by  530 
thousand. 

37.  Multiply  102  times  700  thousand  and  i  hundred  by 
601  thousand  and  twenty. 

38.  Multiply  74  millions  and  21  thousand  by  5  millions 
and  5  thousand. 

39.  Multiply  31  millions  31  thousand  and  31  by  21 
thousand  and  twenty-one. 

40.  Multiply  2  billions,  2  millions,  2  thousand  and  2 
hundred  by  200  thousand  and  2  hundred. 

CASE    IV. 

59.  To  multiply  13,  14,  15,  op  I  with  a  Significant  Figure 
annexed. 

41.  If  one  city  lot  costs  I3245,  what  will  17  lots  cost? 

Analysis. — 17  lots  will  cost  17  times  as  much  as  3245  x  17 
I  lot.  Placing  the  multiplier  on  the  right,  we 
multiply  the  multiplicand  by  the  7  units,  set  each 
figure  one  place  to  the  right  of  the  figure  multi- 
plied, and  add  the  partial  product  to  the  multi- 
plicand.    The  result  is  $55165.     Hence,  the 

EuLE. — I.  Multiply  the  multiplicand  hy  the  iinits^  figure 
of  the  multiplier,  and  set  each  figure  of  the  partial  product 
one  place  to  the  right  of  the  figure  multiplied. 

II.  Add  this  partial  product  to  the  multipflicandyand  the 
result  tvill  be  the  true  product. 

Note.— This  contraction  depends  upon  the  principle  that  as  the 
tens'  figure  of  the  multiplier  is  i,  the  multiplicand  is  the  second  par- 
tial jjroduct ;  hence  its  first  figure  must  stand  in  tejis'  place. 

42.  Multiply  1368  by  13.  43.  Multiply  2106  by  14. 
44.  Multiply  3065  by  15.  45,  Multiply  6742  by  16. 
46.  Multiply  25269  by  18.        47.  Multiply  83467  by  19. 

t;o.  How  mnltiply  hy  ii,  14,  n;,  or  i  with  a  significant  figure  annexed?  Note, 
Upon  what  is  this  contraction  haeed  T 


PIVISION. 

60.  Division  is  finding  how  many  times  one  numh«=i 
is  contained  in  another. 

The  Dividend  is  the  number  to  be  divided. 

The  Divisor  is  the  number  by  which  we  divide. 

The  Qiiotierit  is  the  number  found  by  division,  and 
shows  how  many  times  the  divisor  is  contained  in  the 
dividend. 

The  Manainder  is  a  part  of  the  dividend  left  after 
division.  Thus,  when  it  is  said,  5  is  contained  in  1 7,  3 
times  and  2  over,  17  is  the  dividond,  5  the  divisor,  3  the 
q utient,  and  2  the  remainder. 

Notes. — i.  The  term  dicision,  is  from  tlie  Latin  divido,  to  part, 
or  divide. 

The  term  dididsnd,  from  the  same  root,  signifies  that  which  is  to  he 
divided ;  the  termination  nd,  having  the  force  of  to  he.     (Art.  33,  n.) 

Quotient  is  from  the  Latin  quoties,  signifying  how  often  or  how 
many  times. 

2.  The  remairide?'  is  always  the  same  denomination  as  the  divi- 
dend ;  for  it  is  an  undivided  part  of  it. 

3.  A  proper  remainder  is  always  less  than  the  divisor. 

61.  An  obvious  way  to  find  how  many  times  the  divisor 
is  contained  in  the  dividend,  is  to  subtract  the  divisor 
from  the  dividend  continually,  till  the  latter  is  exhausted, 
or  till  the  remainder  is  less  than  the  divisor;  the  numler 
of  subtractions  will  be  the  quotient.  Thus,  taking  any 
two  numbers,  as  4  and  12,  we  hav3  12  —  4=8;  8  —  4  =  4; 
and  4—4  =  0.  Here  are  three  subtractions;  therefore,  4  is 
contained  in  12,  3  times.     Hence, 

Division  is  sometimes  said  to  be  a  sliort  method  of  con^ 
tinned  subtraction. 


60.  What  is  diviBioTi  ?  The  number  to  he  divided  called  ?  To  divide  by!  The 
remit?  The  part  left?  JV?)fe.  Meanirifr  of  the  term  division  ?  Dividend?  Quo- 
tient ?    Wliat  denomination  is  the  remainder  ?    Why  ? 


54 


DIVISIOK. 


But  there  is  a  shorter  and  more  direct  way  of  obtaining 
the  quotient.  For  we  know  by  the  multiplication  table, 
that  3  times  4,  or  4  taken  3  times,  are  1 2 ;  hence,  4  is 
contained  in  12,  3  times. 

62.  Division  is  the  reverse  of  multiplication.     In  mul- 

tipIicatioQ  both  factors  are  given,  and  it  is  required  to  find 

the  iwoduct ;  in  division,  one  factor  and  the  ])roduct  (which 

answers  to  the  dividend)  are  given,  and  it  is  required  to 

find   the   other  factor,  which   answers   to   the   quotient. 

Hence,  division  may  be  said  to  hQ  finding  a  quotient  which, 

WMltiplied  into  the  divisor,  will  jjrodnce  the  dividend. 

Note. — When  tlie  dividend  contains  only  one  denomination,  tlio 
operation  is  called  Simple  Dicidon. 


DIVISION    TABLE. 

I 

is  in 

2  is  in 

3  is  in 

4  is 

in 

5  is 

in 

6  is  in 

I, 

once. 

2,  once. 

3,  once. 

4,  once. 

5,  once. 

6,  once. 

2, 

2 

4,          2 

6, 

8, 

2 

,10, 

2 

12,        2 

3, 

3 

6,          3 

9,          3 

12, 

^ 

15,. 

^ 

18,        3 

4, 

4 

8,          4 

12,          /j 

16, 

A 

J20, 

^ 

^24,      4 

5, 

5 

10,          5 

15,          5 

20, 

5 

|25, 

q 

30,      5 

6, 

6 

12,          6 

18,          6 

24, 

6 

I30, 

e 

36,        6 

7, 

7 

14,          7 

21,          7 

28, 

7 

35, 

7 

42,        7 

8, 

8 

16,          8 

24,          8 

32, 

8 

40, 

8 

48,        8 

9. 

9 

18,          9 

27,          9 

2>^, 

9 

45, 

9 

54,        9 

10, 

IC 

20,        10 

30,         10 

40, 

10 

j5o,__ 

10 

|6o,       10 

7 

is5  in 

8  is  in 

9  is  in 

10  is 

inl 

II  is 

in 

12  is  in 

7, 

once. 

8,  once. 

9,  once. 

10,  once.  1 

ii,once.| 

1 2  once. 

14, 

2 

16,          2 

18,        2 

20, 

2 

22, 

2 

24,       2 

21, 

3 

24,          3 

27,        3 

30, 

3 

2>2>, 

3 

36,        3 

28, 

4 

32,          4 

36,        4 

40, 

4 

44, 

4 

48,        4 

35, 

5 

40,          5 

45,        5 

50, 

5 

55, 

5 

60,        5 

42, 

6 

48,          6 

54,        6 

60, 

6 

66, 

6 

72,        6 

49, 

7 

56,          7 

63,        7 

70, 

7 

77, 

7 

84,        7 

56, 

8 

64,          8 

72,        8 

80, 

8 

88, 

8 

96,        8 

63, 

9 

72,          9 

81,        9 

90, 

9 

99, 

9 

ro8,        9 

70, 

10 

80,        10 

90,      10 

100, 

10 

no, 

10 

120,      10 

61.  What  is  an  obvions  way  to  find  how  many  times  the  divisor  is  contained 
In  tlie  dividend  ?    WTiat  is  division  sometimes  called  ? 


DIVISIOIS.  DD 

OBJECTS     OF    DIVISION. 
63.  The  object  or  office  of  Division  is  twofold : 
ist.  To  find  how  many  times  one  7iumber  is  contained  in 
another. 

2d,  To  divide  a  niimher  itito  equal  yarts. 

63,  «•  To  find  how  ^uantj  fimes  one  number  is  contained 
in  another. 

1.  A  man  has  15  dollars  to  lay  out  in  books,  which  are 
3  dollars  apiece :  how  many  can  he  buy  ? 

Analysis. — In  this  problem  the  object  is  to  find  how  many  times 
3  dols.  are  contained  in  15  dols. 

Let  the  15  dols.  be  represented  by  15  counters,  or  unit  marks. 
Separating  these  into  groups  of  3  each,  there  are  5  groups.  There- 
fore, he  can  buy  5  books. 

<'^o|<>oo|<><><>|<><><>|<><^<^ 

63,  l>»      To  divide  a  number  into  equal  parts. 

2.  If  a  man  divides  15  dollars  equally  among  3  persons, 
how  many  dollars  will  each  receive  ? 

Analysis. — The  object  here  is  to  divide  15  dols.  into  3  equal  parts. 

Let  the  15  dols.  be  represented  by  15  counters.  If  we  form  3  j^roups, 
first  putting  i  counter  in  each,  then  another,  till  the  counters  are  ex- 
hausted, each  group  will  have  5  counters.  Therefore,  each  person 
will  receive  5  dollars. 

<>0^0<>|<><>0<><C>|00<><>^ 

Remark. — The  preceding  are  representative  examples  of  the  two 
classes  of  problems  to  which  Division  is  applied.  In  the  first,  the 
divisor  and  dividend  are  the  same  denomination,  and  the  quotient  is 
times,  or  an  abstract  number. 

In  the  second,  the  divisor  and  dividend  are  different  denominations, 
and  the  quotient  is  the  same  denomination  as  the  dividend.    Hence, 

64.  When  the  divisor  and  dividend  are  the  same  denom- 
ination, the  quotient  is  always  an  abstract  number. 

62.  Of  what  is  division  the  reverse  ?  What  is  given  in  multiplication  ?  What 
required  ?  What  is  given  in  division  ?  What  required  ?  Note.  When  the  divi- 
dend contains  but  one  denomination,  what  is  the  operation  called  ?  63.  What  is 
the  object  or  oflace  of  division  ?  63,  a.  What  is  the  object  in  the  first  problem? 
63,  b.  What  in  the  second  ?    Rem.  What  is  said  of  first  two  problems  ? 


j:  ly  isi  ON. 

When  the  divisor  and  dividend  are  different  denomina- 
tions, the  quotient  is  always  the  sa7ne  denomination  as  the 
dividend. 

Notes. — i.  The  process  of  separating  a  number  into  equal  parts,  as 
required  in  tlie  second  class  of  problems,  gave  rise  to  the  name 
"  Division."     It  is  also  tlie  origin  of  Fractions.    (Art.  134.) 

2.  The  mode  of  reasoning  in  the  solution  of  these  two  classes  cf 
examples  is  somewhat  different ;  but  the  practical  operation  is  the 
same,  viz. :  to  find  how  many  times  one  number  is  contained  in 
another,  which  accords  with  the  definition  of  Division. 

65.  When  a  number  or  thing  is  divided  into  tivo  equal 
parts,  the  parts  are  called  halves;  into  three,  the  parts  are 
called  thirds  ;  into  four,  they  are  called  fourths  ;  etc. 

The  numher  of  parts  is  indicated  by  their  name. 

66.  A  number  is  divided  into  two,  three,  four,  five,  etc., 
equal  parts  by  dividing  it  by  2,  3,  4,  5,  etc.,  respectively. 

3.  What  is  a  half  of  12  ?  A  third  of  15  ?  A  fourth  of 
20  ?    A  fifth  of  35  ?     A  seventh  of  28  ? 

4.  What  is  a  sixth  of  42  ?  A  seventh  of  56  ?  A  ninth 
of  (iT^  ?     An  eighth  of  72  ?     A  twelfth  of  108  ? 

67.  The  Sign  of  Divisioii  is  a  short  horizontal 
line  between  two  dots  (-^),  placed  before  the  divisor. 
Thus,  the  expression  28-^7,  shows  that  28  is  to  be  divided 
by  7,  and  is  read,  "28  aivided  by  7." 

68.  Division  is  also  denoted  by  loriting  the  divisor 
under  the  dividend,  with  a  short  line  between  them. 
Thus,  -2^  is  equivalent  to  28-^-7.  It  is  read,  "28  divided 
by  7,"  or  "28  sevenths." 

69.  Division  is  commonly  distinguished  as  Sho7i  Divis- 
ion and  Long  Division. 

64.  When  the  divisor  and  dividend  are  the  same  denomination,  what  is  the 
quotient  ?  When  different  denominations,  what  ?  65.  When  a  number  is  divider] 
into  two  equal  parts,  what  are  the  parts  called  ?  Into  three  ?  Four  ?  66.  IIow 
divide  a  number  into  2,  3,  4,  etc.,  equal  parts  ?  67.  What  is  the  sisrn  of  division  ? 
What  does  the  expression  35+7  show?    68.  How  else  is  division  denoted? 


SHOET    DIYISIO]^. 

70.  Short  Division  is  the  method  of  dividhig^ 
when  the  results  of  the  several  steps  are  carried  in  tlie 
mind,  and  the  quotient  only  is  set  down. 

71.  To  divide  by  Sliort  Division. 

Ex.  I.  If  apples  are  $3  a  barrel,  how  many  barrels  can 
you  buy  for  I693  ? 

Analysis.— Since  $3  will  buy  i  barrel,  $693         operatio. 
will  buy  as  many  barrels  as  $3  are  contained        ^-?)^6o*^ 

times  in  $693.     Let  the  numbers  be  set  down  as  ^ 

in  the  margin.    Beginning  at  the  left,  we  proceed       H^^^-  231  Mr. 
thus :  3  is  contained  in  6  hundred,  2  hundred  times. 
Set  the  2  in  hundreds'  place,  under  the  figure  divided,  becavse  it  is 
hundreds.    Next,  3  is  contained  in  9  tens,  3  tens  times.     Set  the  3 
in  tens'  place  under  the  figure  divided.     Finally,  3  is  contained  in 
3  units,  I  time.     Set  the  i  in  units'  place.    Ans.  231  barrels. 

Rem. — This  problem  belongs  to  the  ist  class,  the  object  being  to 
find  how  many  times  one  number  is  contained  in  another,  (Art.  63,  a.) 

Solve  the  following  examples  in  tlie  same  manner : 

(2.)  (3.)  (4.)  (5.) 

2)4468  3)3696  4)4848  5)5555 

6.  A  man  having  27543  pounds  of  grapes,  packed  them 
for  market  in  boxes  containing  5  pounds  each :  how  many 
boxes  did  he  fill,  and  how  many  pounds  over? 

Analysis. — He  used  as  many  boxes  as  there  operation. 

are  times  5  pounds  in  27543  pounds.     Let  the        5)27543  pounds, 
numbers  be  set  down  as  in  the  margin.     Since     Quot  i;qo8"boxes 
the  divisor  5  is  not  contained  in  the  first  figure  ^^^^j    p  Q^gj, 

of  the  dividend,  we  find  how  many  times  it  is 
contained  in  the  first  two  figures,  which  is  5  times  and  2  over.  We 
set  the  quotient  figure  5  under  the  right  hand  figure  divided, 
because  it  is  the  same  order  as  that  figure,  and  prefix  the  remainder 
2,  mentally  to  the  next  figure  of  the  dividend,  making  25.  Now  5 
is  in  25,  5  times,  and  no  remainder.  Again,  5  is  not  contained  in  4, 
the  next  figure  of  the  dividend  ;  we  therefore  place  a  cipher  in  the 


^o  DIVISIOIS". 

quotient,  and  prefix  the  4  mentally  to  the  next  figure  of  the  dividend ^ 
as  if  it  were  a  remainder,  making  43.  Finally,  5  is  in  43,  8  times^ 
and  3  over.  He  therefore  filled  5508  boxes,  and  had  3  pounds 
remainder.     Hence,  the 

EuLE.— I.  Place  the  divisor  on  the  left  of  the  dividend, 
and  heginning  at  the  left,  divide  each  figure  hy  it,  setting 
the  result  under  the  figure  divided. 

II.  If  the  divisor  is  not  contained  in  a  figure  of  the  div- 
idend, put  a  cipher  in  the  quotient,  and  find  hoio  many 
times  it  is  contained  in  this  and  the  next  figure,  setting  the 
remit  under  the  right  hand  figure  divided. 

III.  If  a  remainder  arise  from  any  figure  before  the  last, 
prefix  it  mentally  to  the  next  figure,  and  divide  as  before. 

If  from  the  last  figure,  place  it  over  the  divisor,  and 
annex  it  to  the  quotient. 

Notes. — i.  In  the  operation,  the  divisor  is  placed  on  the  left  of 
tne  dividend,  and  the  quotient  iinder  it,  as  a  matter  of  convenience. 
When  division  is  simply  represented,  the  divisor  is  either 
placed  under  the  dividend,  or  on  the  right,  with  the  sign  (-5-)  be- 
fore it. 

2  The  reason  for  beginning  to  divide  at  the  left  hand  is,  that  in 
dividing  a  higher  order  there  may  be  a  remainder,  which  must  be 
prefixed  to  the  next  lower  order,  as  we  proceed  in  tlie  operation. 

3.  We  place  the  quotient  figure  under  the  figure  divided ;  because 
the  former  is  the  same  order  as  the  latter.    (Art.  71.) 

4.  When  the  divisor  is  not  contained  in  a  figure  of  the  dividend, 
we  place  a  cipher  in  the  quotient,  to  show  that  the  quotient  has  no 
units  corresponding  with  the  order  of  this  figure.  It  also  preserves 
the  local  value  of  the  subsequent  figures  of  the  quotient. 

5.  The  final  remainder  shows  that  a  part  of  the  dividend  is  not 
divided.  It  is  placed  over  the  divisor  and  annexed  to  the  quotient 
to  complete  the  division. 

70.  What  is  Short  Division?  71  How  write  nnmtrers  for  short  division? 
The  next  step  ?  When  the  divisor  is  not  contained  in  a  figure  of  the  dividend, 
how  proceed  ?  When  there  is  a  remainder  after  dividing  a  figure,  how  ?  If  there  is 
a  remainder  after  dividing  the  last  figure,  what?  Note.  Why  place  the  divisor  on 
the  left  of  the  dividend  ?  Why  begin  to  divide  at  the  left  hand  ?  Why  place  each 
quotient  figure  under  the  figure  divided?  Wliy  place  a  cipher  in  the  quotient, 
whin  the  divisor  is  not  contained  in  a  figure  of  the  dividend  ?  What  does  the  final 
remainder  show?    Why  place  it  over  the  divisor  and  annex  it  to  the  Quotient? 


DIVISION.  59 

The  pupil  should  early  learn  to  abbreviate  the  language  used 
In  tlie  process  of  dividing.  Thus,  in  the  next  example,  instead  of 
Baying  5  is  contained  in  7  once,  and  2  over,  let  him  pronounce  the 
quotient  figures  only ;  as,  one,  four,  jive,  seven. 

I.  A  man  divided  7285  acres  of  land  equally  among  his 

5  sons :  what  part,  and  how  much,  did  each  receive  ? 

Solution. — i  is  i  fifth  of  5  ;  hence  each  had  i  fifth  part.  (Art.  65.) 
^gain,  7285  A-7-5  =  i457  A;  hence  each  received  1457  A.    (Art.  66.) 

(2.)  (3.)  (4.)  (5.) 

O436784       3)560346       4)689034       5)748239 

(6.)  (7.)         (8.)  (9.) 

^)3972647     7)4806108     8)7390464     9)8306729 

(10.)  (II.)  (12.) 

T  0)57623140         1 1)667301451  1 2)8160252397 

13.  At  $2  apiece,  liow  many  hats  can  be  bought  for 
$16486  ?     (Art  48,  Note  3.) 

14.  At  $4  a  head,  how  many  sheep  can  be  bought  for 
$844? 

15.  How  many  times  are  3  rods  contained  in  26936 
rods  ? 

16.  A  man  having  $42684,  divided  it  equally  among 
his  4  children :  how  much  did  each  receive  ? 

17.  If  a  quantity  of  muslin  containing  366  yards  is  di- 
vided into  3  equal  parts,  how  many  yards  will  each  part 
contain  ? 

18.  If  84844  pounds  of  bread  are  divided  equally  among 

6  regiments,  how  many  pounds  will  each  regiment  re- 
ceive ? 

19.  Eight  men  found  a  purse  containing  $64968,  which 
they  shared  equally:  how  much  did  each  receive? 


CO  DIVISIOIS^. 

20.  Divide  4268410  by  4.  21.  Divide  5601234  by  6. 

22.  Divide  6403021  by  5.  23.  Divide  7008134  by  7. 

24.  Divide  8210042  by  11.  25.  Divide  9603048  by  8. 

26.  Divide  23468420  by  10.  27.  Divide  32064258  by  9. 

28.  Divide  46785142  by  8.  29.  Divide  59130628  by  7. 

30.  Divide  653000638  by  11.  31.  Divide  774230029  by  12 

32.  In  7  days  there  is  i  week:  bow  many  wrecks  in 
26^6^  days? 

33.  If  $38472  are  divided  equally  among  6  persons,  how 
much  will  each  receive  ? 

34.  How  many  tons  of  coal,  at  87  a  ton,  can  be  pur- 
chased for  I63456? 

35.  At  $9  a  barrel,  how  much  flour  can  be  bought  for 
$47239? 

36.  In  12  months  there  is  i  year:  how  many  years  in 
41260  months? 

37.  A  merchant  laid  out  $45285  in  cloths,  at  $7  a  yard: 
how  many  yards  did  he  buy  ? 

38.  At  8  shillings  to  a  dollar,  how  many  dollars  are 
there  in  75240  shillings? 

39.  In  9  square  feet  there  is  i  square  yard:  how  many 
square  yards  are  there  in  52308  square  feet? 

40.  If  a  person  travel  10  miles  an  hour,  how  long  will 
it  take  him  to  travel  25000  miles? 

41.  A  market  woman  having  845280  eggs,  wished  to 
pack  them  in  baskets  holding  i  dozen  each :  how  many 
baskets  did  it  take  ? 

42.  If  a  prize  of  |i  16248  is  divided  equally  among  8 
men,  what  will  be  each  one's  share  ? 

43.  If  in  7  townships  there  are  2346281  acres,  how  many 
acres  are  there  to  a  tow^nship  ? 

44.  At  $8  a  barrel,  how  many  barrels  of  sugar  can  be 
bought  for  $111364? 

45.  At  $1 1  each,  how  many  cows  can  be  had  for  $88990  ? 


LOITG    DIYISIOH". 

72.  Long  jyivision  is  the  method  of  dividing,  when 
the  results  of  the  several  steps  and  the  quotierit  are  both 
set  down. 

73.  To  divide  by  Long  Division. 

I.  A  speculator  paid  $31097  for  15  city  lots:  what  did 
the  lots  cost  apiece  ? 

Analysis.— Since  15  lots  cost  $31097,  i  operation. 

lot  will  cost  as  many  dollars  as  15  is  con-  I5)$3i097($2073y^ 
tained  times  in  31097.    Let  the  divisor  and  30  '  " 

dividend  be  set  down  as  in  the  margin.  — 

Beginning  at  the  left,  the  first  step  is  to  jog 

find  how  many  times  the  divisor   15,   is  105 

contained  in  31  (which  is  2  times),  and  set  

the  2  on  the  right  of  the  dividend.  ^y 

Second,   multiply   the    divisor    by   the  ac 

quotient  figure,  and  set  the  product   30,  

under  the  figures  divided.     Third,  sub-  2 

tract    this     product    from    the     figures 

divided.  Fourth^  bring  down  the  next  figure  of  the  dividend, 
and  place  it  on  the  right  of  the  remainder,  making  10  for  the 
next  partial  dividend,  and  proceed  as  before.  But  15  is  not  con- 
tained in  10 ;  we  therefore  place  a  cijjher  in  the  quotient,  and 
bring  down  the  next  figure  of  the  dividend,  making  109.  Now 
15  is  in  109,  7  times.  Set  the  quotient  figure  7  on  the  right,  mul- 
tiply the  divisor  by  it,  subtract  the  product  from  the  partial  divi- 
dend, and  to  the  right  of  the  remainder,  bring  down  the  succeeding 
figure  for  the  next  partial  dividend,  precisely  as  before.  Now  1 5  is 
in  47,  3  times.  Setting  the  3  in  the  quotient,  multiplying,  and  sub- 
tracting, as  above,  the  final  remainder  is  2.    We  place  this  remainder 

72.  What  is  Long  Division  ?  73.  How  write  the  numbers  ?  What  is  the  first 
stop?  The  second?  The  third?  The  fourth?  If  the  divisor  is  not  contained 
in  a  partial  dividend,  how  proceed  ?  What  is  to  be  done  with  the  last  reinain-ler  ? 
Note.  What  is  the  difference  between  short  and  long  division  ?  Of  what  order  is 
the  quotient  figure  ? 


62  DIVISION. 

over  the  divisor  and  annex  it  to  the  quotient.  The  divisor  and  divi- 
dend being  different  denominations,  the  quotient  is  the  same  as  the 
dividend  (Art.  64).  Therefore  the  lots  cost  $2073-,%  apiece.   Hence,  the 

EuLE.— I.  Find  liow  many  times  the  divisor  is  contained 
in  the  fewest  figures  on  the  left  of  the  dividend,  that  will 
contain  it,  and  set  the  quotient  on  the  right. 

II.  Multiply  the  divisor  hy  this  quotient  figure,  and  sul- 
tract  the  product  from  the  figures  divided. 

III.  To  the  right  of  the  remainder,  Iring  down  the  next 
figure  of  the  dividend,  and  divide  as  before. 

IV.  If  the  divisor  is  7iot  contained  in  a  partial  dividend, 
place  a  cipher  in  the  quotient,  bring  down  another  figure, 
and  continue  the  operation  till  all  the  figures  are  divided. 

If  there  is  a  remainder  after  dividi7ig  the  last  figure,  set 
it  over  the  divisor,  and  annex  it  to  the  quotie7it. 

Notes. — i.  The  parts  into  which  the  dividend  is  separated  in 
finding  the  quotient  figure,  are  called  partial  dividends,  because 
they  are  parts  of  the  whole  dividend, 

2.  Short  and  Long  Division,  it  will  be  seen,  are  the  same  in  prin- 
ciple. The  only  difference  is,  that  in  one  the  results  of  the  several 
steps  are  carried  in  the  mind,  in  the  other  they  are  set  down. 

Short  Division  is  the  more  expeditious,  and  should  be  employed 
when  the  divisor  does  not  exceed  12. 

The  reasons  for  the  arrangement  of  the  parts,  and  for  beginning 
to  divide  at  the  left  hand,  are  the  same  as  in  Short  Division. 

3.  The  quotient  figure  in  Long  as  well  as  in  Short  Division,  is 
always  of  the  same  order  as  that  of  the  right  hand  figure  of  the  par- 
tial dividend. 

4.  To  prevent  mistakes,  it  is  customary  to  place  a  mark  under 
the  several  figures  of  the  dividend  as  they  are  brought  down. 

5.  After  the  first  quotient  figure  is  obtained,  for  each  succeeding 
figure  of  the  dividend,  either  a  significant  figure  or  a  cipher  must  be 
put  in  the  quotient. 

6.  If  the  product  of  the  divisor  into  the  figure  placed  in  the  quo- 
Uent  is  greater  than  the  partial  dividend,  it  is  plain  the  quotient 
figure  is  too  large,  and  therefore  must  be  diminished. 

If  the  remainder  is  equal  to  or  greater  than  the  divisor,  the  quo- 
tient figure  is  too  sm-cUl,  and  must  be  increased. 


DIVISION.  63 


PROOF. 

74.  By  Multiplication. — Multiply  tlie  divisor  and  quo^ 
tient  together,  and  to  the  product  add  the  remainder.  If 
the  result  is  equal  to  the  dividend,  the  work  is  right. 

Note. — This  proof  depends  upon  the  principle,  that  Division 
is  the  reverse  of  Multiplication  ;  the  dividend  answering  to  the  pro- 
duct, the  divisor  to  one  of  the  factors,  and  the  quotient  to  the  other. 
(Art.  62.) 

75.  By  excess  of  9s. — Multiply  the  excess  of  95  in 
the  divisor  hy  that  in  the  quotient,  and  to  the  product  add 
the  remainder.  If  the  excess  of  gs  i?i  this  sum  is  equal  to 
that  in  the  dividend,  the  work  is  right. 


Hvide  1 8 1403  by  67,  and  prove  the  operation. 

A?is.  270711. 


Proof. — By  Multiplication.— 2707  x  67  =  181369,  and  181369  +  34 
the  rem.  =  181403  the  dividend. 

By  excess  of  9s. — The  excess  of  9s  in  the  divisor  is  4,  and  the 
excess  in  the  quotient  is  7.  Now  4x7=28,  and  28  +  34=62;  the 
excess  of  93  in  62  is  8.     The  excess  of  9s  in  the  dividend  is  also  8. 

3.  Divido  34685  by  15.  4.  Divide  65456  by  16. 

5.  Divide  41534  by  20.  6.  Divide  52663  by  25. 

7.  Divide  420345  by  39.        8.  Divide  506394  by  47. 

9.  Divide  673406  by  69.       10.  Divide  789408  by  77. 
II.  Divide  4375023  by  86.     12.  Divide  5700429  by  93. 
13.  Divide  6004531  by  59.     14.  Divide  8430905  by  78. 
15.  Divide  7895432  by  89.     16.  Divide  9307108  by  98. 

17.  How  many  acres  of  land  at  $75  per  acre,  can  I  buy 
for  I18246? 

18.  At  $83  apiece,  how  many  ambulances  can  be  bought 
for  $37682? 

73.  N^ate.  If  the  product  of  the  divisor  into  the  figure  placed  in  the  quotient,  is 
greater  than  the  partial  dividend,  what  does  it  show  f  If  the  remainder  is  equal 
to  or  greater  than  the  divisor,  what  ?    How  is  Division  proved  ? 


64  Divisioif. 


76.  To  find  the  Quotient  Figure,  when  the  Divisor  is  large. 

Take  i\\Q  first  figure  of  the  divisor  for  a  trial  divisor,  and 
find  hovv'  many  times  it  is  contained  in  the  first  or  first 
tivo  figures  of  the  dividend,  making  due  allowance  for 
carrying  the  tens  of  the  product  of  the  sec07id  figure  of  the 
divisor  into  the  quotient  figure. 

19.  Divide  18046  by  673. 

Analysis. — Taking  6  for  a  trial  divisor,  it  is      673)18046(26 

contained  in  18,  3  times.    But  in  multiplying  7  by  i  ^45 

3,  there  are  2  to  carry,  and  2  added  to  3  times  6,  . 

make  20.     But  20  is  larger  than  the  partial  divi  4^86 

dend  18  ;  therefore,  3  is  too  large  for  the  quotient  40 -^jS 

figure.     Hence,  we  place  2  in  the  quotient,  and 

proceed  as  before.     (Art.  73,  n.)  e  ,g 

20.  Divide  3784123  by  127.     21.  Divide  4361729  by  219. 
22.  Divide  8953046  by  378.     23.  Divide  9073219  by  738. 

24.  How  many  shawls  at  I95,  can  be  bought  for  $42750  ? 

25.  In  144  square  inches  there  is  i  square  foot:  how 
many  square  feet  are  there  in  59264  square  inches  ? 

26.  A  quartermaster  paid  829328  for  312  cavalry  horses: 
how  much  was  that  apiece  ? 

27.  If  128  cubic  feet  of  wood  make  i  cord,  how  many 
cords  are  in  69240  cubic  feet  ? 

28.  If  a  purse  of  $150648  is  divided  equally  among  250 
sailors,  how  much  will  each  receive  ? 

29.  In  1728  cubic  inches  there  is  i  cubic  foot:  how 
many  cubic  feet  are  there  in  250342  cubic  inches  ? 

30.  If  560245  pounds  of  bread  are  divided  equally 
among  11 200  soldiers,  how  much  will  each  receive? 

31.  Div.  36942536  by  4204.       32.  Div."573oo652by5i29. 
33.  Div.  629348206  by  52312.    34.  Div.  730500429  by  61^73. 

35.  Divide  7300400029  by  236421. 

7,6.  Divide  8230124037  by  463205. 

37.  Divide  843000329058  by  203963428. 


DIVISION.  C5 

38.  A  stock  company  having  $5000000,  wa^  divided 
into  1250  shares:  what  was  the  value  of  each  share? 

39.  A  raih'oad  478  miles  in  length  cost  $18120000 :  what 
was  the  cost  per  mile  ? 

40.  A  company  of  942  men  purchased  a  tract  of  land 
containing  272090  acres,  which  they  shared  equally:  what 
was  each  man's  share  ? 

41.  A  tax  of  $42368200  was  assessed  equally  upon  5263 
towns :  what  sum  did  each  town  pay  ? 

42.  The  government  distributed  $9900000  bounty  equally 
among  36000  volunteers:  how  much  did  each  receive? 

43.  Since  there  is  i  year  in  525600  minutes,  how  many 
years  are  there  in  105 192000  minutes. 

CONTRACTIONS. 
CASE    I. 

77.  To  divide  by  a  Composite  Number, 

Ex.  I.  A  farmer  having  300  pounds  of  butter,  packed  it 
in  boxes  of  15  pounds  each :  it  is  required  to  find  how 
many  boxes  he  had,  using  the  factors  of  the  divisor. 

Analysis. — 15  =  5  times  3.  Now  if  he  puts 
5  pounds  into  a  box,  it  is  plain  he  would  use 
as  many  boxes  as  there  are  5s  in  300  or  60  operation. 

jive-pound  boxes.     But   3  five-pound  boxes  5)300  lbs, 

make    i  fifteen-pound   box;    hence,  60  five-  3)60  51b.  boxes. 

pound  boxes  will  make  as  many  15  pound    Ji^ig,    20  boxes. 
boxes,  as  3  is  contained  times  in  60,  which  is 
20.     In  the  operation,  we  first  divide  by  one 
of  the  factors  of  15,  and  the  quotient  by  the  other.    Hence,  the 

KuLE. — Divide  the  dividend  hy  one  of  the  factors  of  the 
divisor,  and  the  quotient  thence  arising  by  another  factor, 
and  so  on,  until  all  the  factors  have  ieeji  used.  The  last 
quotie7it  ivill  he  the  one  required. 

77.  When  the  divisor  Is  a  composite  number,  how  proceed  ? 


OG  DIVISIO-N". 

Notes. — i.  This  contraction  is  the  reverse  of  multiplying  by  a 
composite  number.  Hence,  dividing  the  dividend  (which  answers  to 
the  product)  by  the  several  factors  of  one  of  the  numbers  which  pro- 
duced it,  will  evidently  give  the  quotient,  the  other  factor  of  which 
the  dividend  is  composed,     (Art.  56.) 

2.  When  the  divisor  can  be  resolved  into  different  sets  of  factors, 
the  result  will  be  the  same,  whichever  set  is  taken,  and  whatever 
the  order  in  which  they  are  employed.  The  pupil  is  therefore  at 
liberty  to  select  the  set  and  the  order  most  convenient. 

2.  Divide  357  by  21,  using  the  factors. 

3.  If  532  oranges  are  divided  equally  among  28  boys, 
what  part,  and  how  many  will  each  receive  ? 

4.  A  dairyman  packed  805  pounds  of  butter  in  35  jars : 
how  many  pounds  did  he  put  in  a  jar  2 

5.  How  many  companies  in  a  regiment  containing  756 
soldiers,  allowing  61,  soldiers  to  a  company? 

6.  Divide  204  by  12,  using  different  sets  of  factors. 

7.  Divide  368  by  16,  using  different  sets  of  factors. 

8.  Divide  780  by  30,  using  different  sets  of  factors. 

78.  To  find   the    True  Memainder,  when   Factors  of  the 
divisor  are  used. 

Ex.  9. — A  lad  picked  2425  pints  of  chestnuts,  which  he 
wished  to  put  into  bags  containing  64  pints  each:  it  is 
required  to  find  the  number  of  bags  he  could  fill,  and  the 
number  of  pints  over,  or  the  true  remainder. 

Analysis.  —  The    divisor    64 
equals  the  factors  2x8x4,     Di- 

.,.  .     ,       1  ,7  OPERATION. 

vidmg  2425  pmts  by  2,  the  quo-   2)2425 

tient   is    1212   and   i   remainder.      ~ 

But  the  wm^«  of  the  quotient  1212   8)1212  —  1  Ipt.istr. 

are   2   times   as    large   as   pints,  4)^5^ — 4j   4^2=        8pt.  adr. 

which,  for  the  sake  of  distinction,  oy -? '  3  x  8  X  2  =:=48pt.  3d  r. 

we  will  call  quarts  ;  and  the  re-  

raainder  i,  is  a  pint,  the  same  as  ^ns.   37  bags,  and  5  7  pt.  ovei; 

the  dividend.     (Art.  60,  n) 

Next,  dividing  1212  quarts  by 
8,  the  quotient  is  151,  and  4   remainder.    But  the  units  of  the 


78  How  is  the  trae  remainder  found  ?  Note.  To  what  is  it  equal  ?  What 
phonld  he  done  with  it  ?  The  object  in  multiplying  each  partial  remainder  by  all 
th3  preceding  divisors  except  its  own  ? 


.    DIVISIOif.  6? 

quotient  151  are  8  times  as  large  as  quarts;  call  them  pecks ;  the 
remainder  4,  wliich  denotes  quarts,  must  be  multiplied  by  the 
preceding  divisor  2,  to  reduce  it  back  to  pints.  Finally,  dividing  151 
pecks  by  4,  the  quotient  is  37,  and  3  remainder.  But  the  units  of 
tlie  vjuotient  37  have  four  times  the  value  of  pecks  ;  call  them  hags  ; 
and  tne  remainder  3,  vi^hich  denotes  pecks,  must  be  multiplied  by 
the  last  divisor  8,  to  reduce  them  back  to  quarts,  and  be  multiplied 
by  2  to  reduce  the  quarts  to  pints.  Having  filled  37  bags,  we  have 
three  partial  remainders,  i  pint,  4  quarts,  and  3  pecks. 

The  licxt  step  is  to  find  the  true  remainder.  We  have  seen  that 
a  unit  0/  tlie  2d  remainder  is  twice  as  large  as  those  of  the  given 
dividend,  which  are  pints  ;  hence,  4  quarts  =  8  pints.  Again,  each 
unit  of  tl.e  3d  remainder  is  8  times  as  large  as  those  of  the  preceding 
dividend,  vv  hich  are  quarts  ,  therefore,  3  pecks  =  3  x  8  or  24  quarts  ; 
and  24  ql  J.  -^  24  x  2  or  48  pts.  The  sum  of  these  partial  remainders, 
I  pt.  +  8  pt.  +  48  pt.  =  57  pts.,  is  the  true  remainder.     Hence,  the 

EuLB. — Multiply  each  partial  remainder  ly  all  the 
divisors  preceding  its  own  ;  the  sum  of  these  results  added 
to  the  first  J  ivill  be  the  true  reynainder. 

Notes. — i.  Multiplying  each  remainder  by  all  the  preceding  di- 
visors except  its  own,  reduces  them  to  units  of  the  same  denomina- 
tion as  the  given  dividend.  Hence,  the  true  remainder  is  equal  to 
the  sum  of  the  partial  remainders  reduced  to  the  same  denomina- 
tion as  the  dividend. 

2.  When  found,  it  should  be  placed  over  the  given  divisor,  and  be 
annexed  to  the  quotient. 

10.  Divide  43271  by  45.  11.  Divide  502378  by  67,. 

12.  Divide  710302  by  72.        13.  Divide  3005263  by  84. 
14.  Divide  634005 11  by  96.    15.  Divide  216300265  by  144. 

CASE    II. 

79.  To  divide  by  10,  100,  1000,  etc. 

Ex.  16.  Divide  2615  by  100. 

Analysis. — Annexing  a  cipher  to  a  figure,  we 
have  seen,  multiplies  it  by  10 ;    conversely,  re-  opeeatioh. 

moving  a  cipher  from  the  right  of  a  number     l[oo)26  15 
must   diminish  its  value  10  times,  or  divide  it    Ans.  26  15  Rem. 
by  10 ;  for,  each  figure '  in  the   number  is  re- 
paoved  one  place  to  the  right.     (Art.  12,)    In  like  manner,  cutting 


G8  DIVISIO^-. 

off  two  figures  from  the  right,  divides  it  bj  loo ;  cutting  off  three, 
divides  it  by  looo,  etc. 

In  the  operation,  as  the  divisor  is  loo,  v^^e  simply  cut  off  two 
figures  on  the  right  of  the  dividend ;  the  number  left,  viz.,  26,  is  the 
quotient;  and  the  15  cut  off,  the  remainder.     Hence,  the 

Rule. — From  the  right  of  the  dividend,  cut  off  as  many 
figures  as  there  are  ciphers  in  the  divisor.  The  figures  left 
will  he  the  quotient ;  those  cut  off,  the  remainder. 

17.  Divide  75236  by  100.  20.  9820341  by  looooo. 

18.  Divide  245065  by  1000.        21.  9526401  by  loooooo. 
7:9.  Divide  8052 11  by  loooo.      22.  80043264  by  looooooo. 

CASE    III. 

80.  To  divide, when  the  Divisor  has  Ciphers  on  the  right, 

2^.  At  $30  a  barrel,  how  many  barrels  of  beef  can  be 
bought  for  I4273? 

Analysis. — The  divisor  30  is  composed  of 
the  factors  3  and  10.     Hence,  in  the  operation,  operatioti. 

we  first  divide  by  10,  by  cutting  off  the  right-         3|Q;4   7j3 
hand  figure  of  the  dividend  ;  then  dividing  the     Quot.    142,  i  Eem. 
remaining  figures  of  the  dividend  by  3,  the       A71S.  142] ^  Us. 
quotient  is  142,  and  i  remainder.     Prefixing 

the  I  remainder  to  the  3  which  was  cut  off,  we  have  13  for  the  true 
remainder,  which  being  placed  over  the  given  divisor  30,  and  an- 
nexed to  the  quotient,  gives  142 3^  bis.  for  the  answer.     Hence,  the 

Rule. — I.  Cut  off  the  ciphers  on  the  iHght  of  the  divisor 
and  as  many  figures  on  the  right  of  the  dividend. 

II.  Divide  the  remaining  part  of  the  dividend  ly  the 
remaining  part  of  the  divisor  for  the  quotient. 

III.  Annex  the  figures  cut  off  to  the  remainder,  and  the 
result  will  he  the  true  remainder.     (Art.  78.) 

Note.— This  contraction  is  based  upon  the  last  two  Cases.  The 
true  remainder  should  be  placed  over  the  ichole  divisor,  and  be  an- 
nexed to  the  quotient. 

70.  ITow  proceed  when  the  divisor  is  10,  100,  etc.  ?  80.  How  when  there  ar/ 
ciphers  on  the  ri^ht  of  the  divisor?    Note.  Upon  what  is  this  contraction  based' 


DiYisiOiq^.  G9 

(4.  Divide  45678  by  20.  25.  81386  by  200. 

26.  Divide  603245  by  3400.  27.  74032 1  by  6500. 

28.  Divide  7341264  by  87000.        29.  8004367  by  93000. 
30.  Divide  61273203  by  125000.     31.  416043271  by  67000a 

32.  In  100  cents  tbere  is  i  dollar:  how  many  dollars  in 
37300  cents  ? 

S3.  At  $200  apiece,  how  many  horses  will  $45800  buy? 

34.  If  $75360  were  equally  distributed  among  1000  men, 
how  much  would  each  receive  ? 

35.  At  I4800  a  lot,  how  many  can  be  had  for  I25200  ? 

36.  How  many  bales,  weighing  450  pounds  each,  can  be 
made  of  27000  pounds  of  cotton  ? 


QUESTIONS    FOR    REVIEW,    INVOLVING    THE 
PRECEDING    RULES. 

1.  William,  who  has  219  marbles,  has   73  more  than 
James:  how  many  has  James  ?     How  many  have  both  ? 

2.  A  farmer  having  $6S  sheep,  wishes  to  increase  his 
flock  to  775  :  how  many  must  he  buy? 

3.  The  diflFerence  of  two  persons'  ages  is  19  years,  and 
the  younger  is  5  7  years :  what  is  the  age  of  the  elder  ? 

4.  What  number  must  be  added  to  1368  to  make  3147  ? 

5.  What  number  subtracted  from  41 18  leaves  1025  ? 

6.  What  number  multiplied  by  95  will  produce  7905  ? 

7.  The  product  of  the  length  into  the  breadth  of  a  field 
is  2967  rods, and  the  length  is  69  rods:  what  is  the  breadth? 

8.  A  man  having  5263  bushels  of  grain,  sold  all  but 
145  bushels:  how  much  did  he  sell? 

9.  What   number  must  be  divided  by   87,  that  the 
quotient  may  be  99  ? 

10.  If  the  quotient  is  217,  and  the  dividend  7595,  what 
must  be  the  divisor  ? 

11.  If  the  divisor  is  341  and  the  quotient  589,  what 
must  be  the  dividend  ? 


70  DIVISION. 

12.  A  merchant  bought  516  barrels  of  flour  at  $g  a  bar- 
rel, and  sold  it  for  $5275  :  how  much  did  he  gain  or  lose  ? 

13.  How  long  can  250  men  subsist  on  a  quantity  of 
food  sufficient  to  last  i  man  7550  days? 

14.  How  many  pounds  of  sugar,  at  11  cents,  must  be 
given  for  629  pounds  of  coffee,  at  17  cents  ? 

15.  At  $13  a  barrel,  how  many  barrels  of  flour  must  be 
given  for  530  barrels  of  potatoes  worth  $3  a  barrel  ? 

16.  A  man  having  $15260,  deducted  $4500  for  personal 
use,  and  divided  the  balance  equally  among  his  7  sons : 
how  much  did  each  son  receive  ? 

17.  A  man  earns  12  shillings  a  day,  and  his  son  8  shil- 
lings: how  long  will  it  take  both  to  earn  1200  shilhngs? 

18.  If  a  man  earns  $19  a  week,  and  pays  I2  a  week  fo^ 
boarding  each  of  his  3  sons  at  school,  how  much  will  he 
lay  up  in  1 2  weeks  ? 

19.  A  farmer  sold  6  cows  at  $2;^,  150  bushels  of  wheat 
at  $2,  and  75  barrels  of  apples  at  $4,  and  laid  out  his  money 
in  cloth  at  $7  a  yard:  how  many  yards  did  he  have ? 

20.  If  I  buy  1 36 1  barrels  of  flour  at  $7,  and  sell  the 
whole  for  $12249,  how  much  shall  I  make  per  barrel  ? 

21.  Tlie  earnings  of  a  man  and  his  two  sons  amount  to 
$3560  a  year;  their  expenses  are  $754.  If  the  balance  is 
divided  equally,  what  will  each  have  ? 

22.  A  man  having  $23268,  owed  $1733,  and  divided  the 
rest  aniong  four  charities :  how  much  did  each  receive  ? 

23.  How  many  sheep  at  $5  a  head,  must  be  given  for  30 
cows  at  $42  apiece  ? 

24.  A  father  bought  a  suit  of  clothes  for  each  of  his 
3  sons,  at  $123  a  suit,  and  agreed  to  pay  17  tons  of  hay  at 
$12  a  ton,  and  the  rest  in  potatoes  at  $4  a  barrel:  how 
many  barrels  of  potatoes  did  it  take  ? 

25.  If  you  add  $435  to  $567,  divide  the  sum  by  $334, 
multiply  the  quotient  by  217,  and  divide  the  product  by 
59,  what  will  be  the  result? 


Diyisioi^^.  71 

26.  If  from  1530  you  take  319,  add  793  to  the  remainder, 
multiply  the  sum  by  44,  and  divide  the  product  by  37, 
what  will  be  the  result  ? 

27.  A  man's  annual  income  is  $4250;  if  he  spends  $1365 
for  house  rent,  $1439  ^r  other  expenses,  and  the  balance 
in  books,  at  $3  apiece,  how  many  books  can  he  buy  ? 

28.  A  farmer  having  $3038,  bought  15  tons  of  hay  at 
$1 1, 3  yoke  of  oxen  at  $155, 375  sheep  at  $5,  and  spent  the 
rest  for  cows  at  $41  a  head :  how  many  cows  did  he  buy  ? 

GENERAL    PRINCIPLES    OF    DIVISION. 

81.  From  the  nature  of  Division,  the  absolute  value  of  the 
quotient  depends  both  upon  the  divisor  and  the  dividend. 

The  relative  value  of  the  quotient ;  that  is,  its  value  com- 
pared with  the  dividend,  depends  upon  the  divisor.    Thus, 

1.  If  the  divisor  is  equal  to  the  dividend,  the  quotient  is  i. 

2.  If  the  divisor  is  greater  than  the  dividend,  the 
quotient  is  less  than  i. 

3.  If  the  divisor  is  less  than  the  dividend,  the  quotient 
i^  greater  than  i. 

4.  If  the  divisor  is  i,  the  quotient  is  equal  to  the  dividend. 

5.  If  the  divisor  is  greater  than  i,  the  quotient  is  les? 
than  the  dividend. 

6.  If  the  divisor  is  less  than  i,  the  quotient  is  greater 
than  the  dividend. 

82.  The  relation  of  the  divisor,  dividend,  and  quotient 
is  such  that  the  divisor  remaining  the  same. 

Multiplying  the  dividend  by  any  number,  multiplies  the 
quotient  by  that  number.  Thus,  12-^2=6;  and  (12x2) 
-7-2=6x2.     Conversely, 

81.  Upon  what  does  the  absolute  vahie  of  the  quotient  depend?  Its  relative 
Talue  ?  If  the  divisor  is  equal  to  the  dividend,  what  is  true  oi  the  quotient  ?  If 
greater?  If  less?  If  the  divisor  is  t,  what  is  the  quotient?  If  £;reater  than  i  ? 
If  less  than  i  f  82.  What  is  the  effect  of  multiplying  the  dividend  ?  83.  Of  dirid- 
Incrit? 


TZ  DIVISION, 

83.  Dividing  tlie  dividend  l>y  any  number,  divides  the 
quotient  by  that  number.     Thus,  (12-^2)-:- 2  =  6h- 2. 

84.  Multiplying  the  divisor  by  any  number,  divides  the 
quotient  by  that  number.  Thus,  24-^-6=4;  and  24-r 
(6x2)  =4~  2.     Conversely, 

85.  Dividi7ig  the  divisor  by  any  number,  multiplies  the 
quotient  by  that  number.     Thus,  24-^  (6-^2) =4  x  2. 

86.  Multi2)lying  or  dividing  both  the  divisor  and  d'^V^- 
Je/it?  by  the  same  number,  does  not  6j//er  the  quotient. 
Thus,  48-^8  =  6;  so  (48x2)4-(8x2)=:6;  and  (48-^2)-t- 
(8-^-2)1:^6. 

Note. — Multiplying  or  dividing  tlie  dividend,  produces  a  like 
effect  on  the  quotient ;  but  multiplying  or  dividing  the  divisor,  pro- 
duces the  opposite  effect  on  the  quotient. 

86,  «•  If  a  number  is  both  multiplied  and  divided  by  the 
same  number,  its  value  is  not  altered.  Thus,  (7  x  6) -=-6  is 
equal  to  7. 


PROBLEMS    AND     FORMULAS     PERTAINING    TO     THE 
FUNDAMENTAL    RULES. 

87.  A  I^rohlem  is  something  to  he  done,  or  a  question 
to  be  solved. 

88.  A  Formula  is  a  specific  rule  by  which  problems 
are  solved,  and  may  be  expressed  by  common  language,  or 
by  signs. 

89.  The  four  great  Problems  of  Arithmetic  have  already 
been  illustrated.     They  are — 

ist.  When  two  or  more  numbers  are  given,  to  find  their 
dum,  or  amount.     (Art.  29.) 

2d.  When  two  numbers  are  given,  to  find  their  difference, 

84.  What  is  the  effect  of  multiplying  the  divipor  ?  Of  dividing  it  ?  86.  WTial 
Ib  the  eflFect  of  multiplying  or  dividing  both  the  divisor  and  dividend  ?  87.  What 
id  a   problem?     8S.   A  formula?     89.  The  four  great  problems  ot  arithmetic  1 


PROBLEMS    AN"D    FORMULAS.  73 

3d.  When  two  factors  are  giyen,  to  find  tlieir  product 
(Art.  50.) 

4tb.  When  two  members  are  given,  to  find  how  man^ 
times  one  is  contained  in  the  other.     (Art.  73.) 

90.  These  problems  constitute  the  four  fundamental 
rules  of  Arithmetic,  called  Addition,  Subtraction,  Multi- 
plication, and  Division. 

Notes. — i.  These  rules  are  called  fundamental,  because  upon 
them  are  based  all  arithmetical  operations. 

2.  As  multij)licatioji  is  an  abbrevated  form  of  addition,  and  division 
of  subtraction,  it  follows  that  every  change  made  upon  the  value  of  a 
number,  must  increase  or  diminish  it.  Hence,  strictly  speaking, 
there  are  but  two  fundamental  operations,  viz. :  aggregation  and 
diminution,  or  increase  and  decrease. 

3.  The  following  problems,  though  subordinate,  are  so  closely  con- 
nected with  the  preceding,  that  a  passing  notice  of  them  may  not  be 
improper,  in  this  connection. 

91.  To  find  the  greater  of  two  numbers,  the  less  and  their 
difference  being  given. 

1.  A  planter  raised  two  successive  crops  of  cotton,  the 
smaller  of  which  amounted  to  4168  bales,  and  the  dif- 
ference between  them  was  11 23  bales:  what  was  the 
greater  crop  ? 

Analysis. — If  the  difference  between  two  numbers  be  added  to 
the  less,  it  is  obvious  the  sum  must  be  equal  to  the  greater.  There- 
fore, 4168  bales  +  1 123  bales  or  5291  bales  must  be  the  greater  crop. 
Hence,  the 

Rule. — To  the  less  add  the  difference,  and  the  sum  will 
be  tlie  greater.     (Art.  39.) 

2.  One  of  the  two  candidates  at  a  certain  election,  re- 
ceived 746  votes,  and  was  defeated  by  a  majority  of  411: 
how  many  votes  did  the  successful  candidate  receive  ? 

90.  What  do  these  constitute?  Note.  Why  so  called?  What  is  given  and 
■what  required  In  addition  ?  In  subtraction  ?  In  multiplication  ?  In  division  ? 
Where  begin  the  operation  in  addition,  subtraction,  and  multiplication  ?  Where 
tn  division  ?  WTiat  is  the  difference  between  addition  and  subtraction  ?  Between 
addition  and  multiplication  ?  Between  subtraction  and  division  ?  Between 
multiplication  and  division  ? 


74  PEOBLEMS    AKD    FORMULAS. 

92.  To  find   the   less  of  two    numbers,  the  greater   and 
their  difference  being  given. 

3.  The  greater  of  two  cargoes  of  flour  is  526.7  barrels, 
and  their  difference  is  1348  barrels:  how  many  barrels  does 
the  smaller  contain  ? 

Analysis. — The  difference  added  to  the  less  number  equals  the 
greater  ;  therefore,  the  greater  diminished  by  the  difference,  must  be 
equal  to  the  less  ;  and  5267  barrels  minus  1348  barrels  leaves  3919 
barrels,  the  smaller  cargo.     Hence,  the 

EuLE. — From  the  greater  subtract  the  difference,  and  the 
result  ivill  he  the  less.     (Art.  38.) 

4.  At  a  certain  election  one  of  the  two  candidates  re- 
ceived 1366  votes,  and  was  elected  by  a  majority  of  219 
votes :  how  many  votes  did  the  other  candidate  receive  ? 

93.  The   Product   and   one    Factor    being  given,  to   find  the 

other  Factor. 

5.  A  drover  being  asked  how  many  animals  he  had, 
replied  that  he  had  67  oxen,  and  if  his  oxen  were  multi- 
plied by  his  number  of  sheep,  the  product  would  be 
37520:  how  many  sheep  had  he? 

Analysis. — Since  3752013  Si  product,  and  67  one  of  its  factors,  the 
other  factor  must  be  as  many  as  there  are  67s  in  37520;  and 
37520-^67=560.  (Art.  93.)    Therefore,  he  had  560  sheep.    Hence,  the 

EuLE. — Divide  the  product  hy  the  given  factor,  and  the 
quotient  will  he  the  factor  required.     (Art.  62.) 

6.  The  length  of  a  certain  park  is  320  rods,  and  the 
product  of  its  length  and  breadth  is  51200  rods:  what  is 
its  breadth  ? 

94.  The   Product    of  three    or    more    Factors    and    all   the 

Factors  but  one  being  given,  to  find  that  Factor. 

7.  The  product  of  the  length,  breadth,  and  height  of 
a  certain  mound  is  62730  feet;  its  length  is  45  feet,  and 
its  breadth  41  feet:  what  is  its  height? 

91.  How  find  the  greater  of  two  numbers,  the  less  and  difference  being  given? 
Q*,  How  find  the  less,  the  greater  and  difference  being  given  ? 


PROBLEMS     Al^^D     FORMULAS.  75 

Analysis. — The  contents  of  solid  bodies  are  found  by  multiplying 
their  length,  breadth,  and  thickness  together.  Now  as  the  length  is 
45  feet,  and  the  breadth  41  feet,  the  product  of  which  is  1845,  the 
height  must  be  62730  feet  divided  by  1845,  or  34  feet,    Hence,  the 

EuLE. — Divide  the  given  product  by  the  product  of 
the  given  factors,  arid  the  quotient  will  le  the  factor  re- 
quired.    (Art,  93.) 

8.  The  continued  product  of  tlie  distances  wh'  h.  4  men 
traveled  is  1944630  miles;  one  traveled  45,  nother  41, 
and  another  34  miles:  how  far  did  the  fourth  travel? 

95.  To  find  the  Dividend,  the   Divisor  and  Quotient  being 

given. 

9.  If  the  quotient  is  7071,  and  the  divisor  556,  what  is 
the  dividend  ? 

Analysis. — Since  the  quotient  shows  how  many  times  the  divisor 
is  contained  in  the  dividend,  it  follows,  that  the  product  of  the 
divisor  and  quotient  must  be  equal  to  the  dividend.  Now  7071  x  556 
=  3931476  the  dividend.     Hence,  the 

Rule. — Multiply  the  divisor  ly  the  quotient,  and  the 
result  luill  he  the  dividend.  (Art.  74.) 

10.  What  number  of  dollars  divided  among  135  persons 
will  give  them  $168  apiece? 

96.  To  find    the    Divisor,   the    Dividend   and    Quotient 

being   given. 

11.  What  must  8640  be  divided  by  that  the  quotient 
may  be  144? 

Analysis. — Since  the  quotient  shows  how  many  times  the  divisor 
is  contained  in  the  dividend,  it  follows  that  if  the  dividend  is  divided 
by  the  quotient  the  result  must  be  the  divisor,  and  8640^144=60. 
Therefore,  the  divisor  is  60.     Hence,  the 

Rule. — Divide  the  dividend  by  th^  quotient,  and  the 
result  will  be  the  divisor.      (Arts.  62,  93.) 

93.  The  product  and  one  factor  being  given,  how  find  the  other  fiactort 


76  PHOBLEMS     AND     FORMULAS. 

12.  If  the  dividend  is  7620,  and  the  quotient  127,  what 

must  be  the  divisor  ? 

97.  The  Sum  and  Difference  of  two  numbers  being  given, 
to  find  the  Nmnbers. 

13.  The  sum  of  two  numbers  is  65,  and  their  difference 
15  :  what  are  the  numbers  ? 

Analysis. — The  sum  65  is  equal  to  the  greater  number  increased 
by  the  less;  and  the  greater  diminished  by  the  difference  15,  is 
equal  to  the  less  (Art.  38).  Hence,  if  15  is  taken  from  65,  the  re- 
mainder 50,  must  be  twice  the  less.  But  50-7-2—25  the  less  num- 
ber ;  and  25  +  15=40  the  greater  number. 

Proof. — 404-25  =  65  the  given  sum.     Hence,  the 

Rule. — From  tlie  sum  take  .the  difference,  and  half  the 
remainder  ivill  be  the  less  number. 

To  the  less  add  the  difference,  and  the  result  ivill  be  the 
greater.     (Art.  39.) 

14.  A  merchant  made  $5368  in  two  years,  and  the  dif- 
ference in  his  annual  gain  was  $976:  what  was  his  profit 
each  year  ? 

15.  The  whole  number  of  votes  cast  for  the  two  candi- 
dates at  a  certain  election  was  5564,  and  the  successful 
candidate  was  elected  by  a  majority  708 :  how  many  votes 
did  each  receive  ? 

16.  A  lady  paid  S250  for  her  watch  and  chain;  the 
former  being  valued  $42  higher  than  the  latter :  what  was 
the  price  of  each  ? 

17.  Two  pupils  A  and  B, solved  75  examples;  B  solving 
15  less  than  A:  how  many  did  each  solve? 

1 3.  A  and  B  found  a  pocket-book,  and  returning  it  to 
the  owner,  received  a  reward  of  $500,  of  which  A  took 
$38  more  than  B :  what  was  the  share  of  each  ? 


94.  When  the  product  of  three  or  more  factors,  and  all  hut  one  are  given,  how 
find  that  one  ?  95.  IIow  find  the  dividend,  the  divisor  and  quotient  being  given? 
96.  How  find  the  divisor,  the  dividend  and  quotient  being  given  ? 


ANALYSIS. 

98.  Analysis  primarily  denotes  the  seioaration  of  an 
object  into  its  elements. 

99.  Analysis,  in  AviihrnQiic,!^  the  process  of  tracing 
the  relation  of  the  conditions  of  a  problem  to  each  other, 
and  thence  deducing  the  successive  stejjs  necessary  for  its 
solution. 

Notes, — i.  The  application  of  Analysis  to  aritlimetic  is  of  recent 
date,  and  to  this  source  the  late  improvements  in  the  mode  of 
teaching  the  subject  are  chi-efly  due.  Previous  to  this,  Arithmetic 
to  most  pupils,  was  a  hidden  mystery,  regarded  as  beyond  the  reach 
of  all  but  the  favored  few. 

2.  The  pupil  has  already  learned  to  analyze  particular  examples, 
and  from  them  to  deduce  specific  rules  by  which  similar  examples 
may  be  solved  ;  but  Arithmetical  Analysis  has  a  much  wider  range. 
It  is  applied  with  advantage  to  those  classes  of  examples  commonly 
placed  under  the  heads  of  Barter,  Percentage,  Profit  and  Loss, 
Simple  and  Compound  Proportion,  Partnership,  etc.  In  a  word,  it 
is  the  grand  Common- Sense  Rule  by  which  business  men  perform 
the  great  majority  of  commercial  calculations. 

100.  1^0  specific  dtrectio7is  can  be  prescribed  for  analyti- 
cal solutions.  The  following  suggestions  may,  howeveic 
be  serviceable  to  beginners : 

ist.  In  general,  we  reason  from  the  given  value  of  one, 
to  the  value  of  two  or  more  of  the  same  kind.    Or, 

2d.  From  the  given  value  of  two  or  more,  to  that  of  one. 

In  the  first  instance  we  reason  from  a  part  to  the 
whole  ;  in  the  second,  from  the  ichole  to  a  yart. 

3d.  Sometimes  the  result  of  certain  combinations  ia 
given,  to  find  the  original  number  or  lase. 

98.  What  is  the  primary  meaning  of  analysis  ?  99.  What  is  arithmetical 
analysis?  Note.  What  is  said  as  to  its  utility  in  arithmetical  and  business 
calculations  ?  100.  Can  specific  rules  be  given  for  analytical  solutions  t  What 
general  directions  can  you  mention  ? 


78  Al^ALTSIS. 

In  such  cases,  it  is  generally  best  to  begin  with  the 
result,  and  reverse  each  operation  in  succession,  till  the 
original  number  is  reached.  That  is,  to  reason  from  the 
result  to  its  origin,  or  from  effect  to  cause. 

1.  A  farmer  bought  150  sheep  at  $2  a  head,  and  paid 
for  them  in  cows  at  $20  a  head :  how  many  cows  did  it 
take  to  pay  for  the  sheep  ? 

2.  How  much  tea,  at  85  cents  a  pound,  must  be  given 
for  425  pounds  of  rice,  at  10  cents  a  pound  ? 

3.  How  much  sugar,  at  1 2  cents  a  pound,  must  be  given 
for  288  pounds  of  raisins,  at  18  cents  a  pound? 

4.  How  much  corn,  at  80  cents  a  bushel,  must  be  given 
for  160  pounds  of  tobacco,  at  30  cents  a  pound? 

5.  How  much  butter,  at  40  cents  a  pound,  must  be 
given  for  62  yards  of  cahco,  at  20  cents  a  yard  ? 

6.  Bought  189  yards  of  linen,  at  84  cents  a  yard,  and 
paid  for  it  in  oats,  at  42  cents  a  bushel :  how  many  bushels 
did  it  take  ? 

7.  Paid  18  barrels  of  flour  for  30  yards  of  cloth,  worth 
$6  a  yard :  what  was  the  flour  a  barrel  ? 

8.  A  farmer  gave  15  loads  of  hay  for  45  tons  of  coal, 
worth  %6  a  ton :  what  did  he  receive  a  load  for  his  hay  ? 

9.  A  sold  B  35  hundred  pounds  of  hops,  at  I27  a  hun- 
dred, and  took  45  sacks  of  coffee,  at  $14  a  sack,  and  the 
balance  in  money :  how  much  money  did  he  receive  ? 

10.  James  bought  96  apples,  at  the  rate  of  4  for  3  cents, 
and  exchanged  them  for  pears  at  4  cents  apiece:  how 
many  pears  did  he  receive  ? 

11.  A  farmer  being  asked  how  many  acres  he  had, 
replied,  if  you  subtract  20  from  the  number,  divide  the 
remainder  by  8,  add  15  to  the  quotient,  and  multiply  by  5, 
the  product  will  be  125  :  how  many  had  he  ? 

Analysis, — Taking  125  as  the  base,  and  reversing  the  several 
operations,  beginning  with  the  last,  we  have  125  -f-  5  =  25,  the 
miraber  before  the  multiplication.  Again,  subtracting  15  from  25. 
we  have  25  —  15  =  10,  the  number  before  the  addition. 


A2^ALYSIS.  79 

Next,  multiplying  lo  by  8,  we  liave  lo  x  8=:8o,  the  number  before 
tlie  division.  Finally,  adding  20  to  80,  we  have  80  +  20=100,  the 
number  required. 

12.  What  number  is  that,  to  which,  if  25  be  added,  and 
the  sum  multiplied  by  9,  the  product  will  be  504? 

13.  What  number,  if  diminished  by  40,  and  the  remain- 
der divided  by  8,  the  quotient  will  be  58  ? 

14.  A  man  being  asked  how  many  children  he  had, 
answered,  if  you  multiply  the  number  by  11,  add  23  to 
the  product,  and  divide  the  sum  by  9,  the  quotient  will 
be  16 :  how  many  children  had  he  ? 

15.  The  greater  of  two  numbers  is  3  times  the  less,  and 
the  sum  of  the  numbers  is  s^ :  what  are  the  numbers  ? 

Analysis.— The  smaller  number  is  i  part,  and  the  larger  3  parts ; 
hence,  the  sum  of  the  two  is  4  parts,  which  by  the  conditions  is  36. 
Now,  if  4  parts  of  a  nijm.ber  are  36,  i  part  is  equal  to  as  many  units 
as  there  are  4s  in  36,  or  9.  Therefore  9  is  the  smaller  number,  and 
3  times  9  or  27,  the  greater. 

16.  The  sum  of  two  numbers  is  72,  and  the  greater  is 
5  times  the  less  •  what  are  the  numbers  ? 

17.  Divide  472  into  three  such  parts,  that  the  second 
shall  be  twice  the  first,  and  the  third  3  times  the  second 
plus  13. 

Analysis. — Calling  the  first  i  part,  the  second  will  be  2  parts, 
and  the  tliird  6  parts  plus  13 ;  hence  the  sum  of  the  three,  in  the 
terms  of  the  first,  is  9  parts  plus  13,  which  by  the  conditions  is  472, 
Taking  13  from  472  leaves  459,  and  we  have  9  parts  equal  to  459. 
Now  if  9  parts  equal  459,  i  part  is  equal  to  as  many  units  as  9  is 
contained  times  in  459,  or  51.  Therefore  the  first  is  51,  the  second 
2  times  51  or  102,  the  third  3  times  102  plus  13  or  319. 

18.  A  and  B  counting  their  money,  found  that  both 
had  $473,  and  that  A  had  3  times  as  much  as  B  plus  $25  ; 
how  much  had  each  ? 

19.  The  sum  of  two  numbers  is  243,  the  second  is 
three  times  the  first  minus  25  :  what  are  the  numbers  ? 

20.  What  number  is  that  to  which  if  315  be  added,  the 
sum  will  be  250  less  than  2683  ? 

(For  further  applications  of  Analysis,  see  subsequent  pages.) 


CLASSIFICATION 

AND    PROPERTIES   OF  NUMBERS. 

101.  Numbers  are  divided  into  abstract  and  concrete, 
simple  and  compound,  prime  and  comp)osite,  odd  and  even, 
integral,  fractional,  and  mixed,  Tcnown  and  unhnoivn, 
similar  and  dissimilar,  commensurable  and  incommen- 
surable, rational  and  irrational  or  surds. 

Def. — I.  Abstract  Numbers  are  those  which  are  not  applied  to 
things ;  as,  one,  two,  three. 

2.  Concrete  Numbers  are  those  which  are  applied  to  things ;  as, 
two  caps,  three  pencils,  six  yards. 

3.  Simple  Numbers  are  those  which  contain  only  one  denomination, 
and  may  be  either  abstract  or  concrete;  as,  13,  11  pounds. 

4.  A  Compound  Number  is  one  containing  two  or  rn^o^e  denamina- 
tions,  which  have  the  same  base  or  nature ;  as,  3  shillings  and  6 
pence ;  4  yards  2  feet  and  6  inches. 

5.  A  Prime  Number  is  one  which  cannot  be  produced  by  multi 
plication  of  any  two  or  more  numbers,  except  a  unit  and  itself. 

All  prime  numbers  except  2  and  5,  end  in  i,  3,  7,  or  9. 

6.  A  Composite  Number  is  the  product  of  two  or  -move  factors,  each 
of  which  is  greater  than  i ;  as,  15  (5  x  3),  24  (2  x  3  x  4),  etc.    (Art.  5£,.) 

A  prime  number  diflfers  from  a  composite  number  in  two  respects :  First,  In 
its  origin ;  Second,  In  its  divisibility  ;  the  former  being  divisible  only  by  a  unit 
and  itself;  the  latter  by  each  of  the  factors  which  produce  it. 

7.  Two  numbers  are  prime  to  each  other  or  relatively  prime,  when 
the  only  nimiber  by  which  both  can  be  divided  without  a  remainder, 
is  a  unit  or  i ;  as,  5  and  6. 

8.  An  Even  Number  is  one  wliich  can  be  divided  by  2  without  a 
remainder;  as,  4,  6,  10. 

9.  An  Odd  Number  is  one  which  cannot  be  divided  by  2  without  a 
remainder;  as,  3,  5,  7,  9,  11. 

Name  the  kind  of  each  of  the  following  numbers,  and  why  :  3,  10, 
17,  21,  28,  31,  56,  63,  72,  81,  44,  39,  91,  67,  51,  84,  99,  100. 

10.  An  Integer  is  a  number  which  contains  one  or  more  entire 
units  only;  as,  i,  3,  7,  10,  50,  100. 

loi.  How  are  numbers  divided  ?  An  abstract  number  ?  Concrete  ?  Simple  ? 
Compound  ?  Prime  ?  Composite  ?  When  are  two  numbers  prime  to  each  other ! 
Eren  ?    Odd  ?    An  integer  ?    A  fraction  ?    A  mixed  number  ? 


PROPERTIES    OF    ]SrUMBERS.  81 

11.  A  Fraction  is  one  or  more  of  the  equal  parts  into  which  a  unit 
is  divided  ;  as,  i-half,  2-thirds,  3-fourths,  etc. 

12.  A  Mixed  Number  is  an  integer  and  a  fraction  expressed 
together  ;  as,  5^^,  iif,  etc. 

13.  Like  or  similar  numbers  are  those  which  express  units  of  the 
same  kind  or  denomination  ;  as,  3  shillings  and  5  shillings,  four  and 
seven,  etc. 

14.  Unlike  Numbers  are  those  which  express  units  of  different 
kinds  or  denominations ;  as,  2  apples  and  3  oranges. 

15.  Commensurable  Numbers  are  tliose  which  can  be  divided  by 
the  same  number,  without  a  remainder ;  as,  9  and  12,  each  of  which 
can  be  divided  by  3. 

16.  Incommensurable  Numbers  are  those  which  cannot  be  divided 
by  the  same  number  without  a  remainder.  Thus,  3  and  7  are  incom- 
mensurable. 

17.  A  given  number  is  one  whose  value  is  expressed. 

A  number  is  also  said  to  be  given  when  it  can  be  easily  inferred 
from  something  else  which  is  given.  Thus,  if  two  numbers  are 
given,  their  sum  and  difference  are  given. 

18.  An  Unknown  Number  is  one  whose  value  is  not  given. 

102.  A  Factor  of  a  number  is  one  of  the  numbers, 
which  multiplied  together,  produce  that  number. 

103.  An  Exact  Divisor  of  a  number  is  one  which 
will  divide  it  without  a  remainder.  Thus  2  is  an  exact 
divisor  of  6,  3  of  15,  etc. 

Notes. — i.  An  exact  divisor  of  a  number  is  always  &  factor  of  that 
number ;  and,  conversely,  a  factor  of  a  number  is  always  an  exact 
divisor  of  it.  For,  the  dividend  is  the  product  of  the  didsor  and 
quotient,  and  therefore  is  divisible  by  each  of  the  numbers  that  pro- 
duce it.     (Art.  62.) 

2.  The  terms  divisor  and  factor  are  here  restricted  to  integral 
numbers. 

104.  A  3Icasii7^e  of  a  number  is  an  exact  divisor  of 
that  number.  It  is  so  called  because  the  cor)iparath)0 
magnitude  of  the  number  divided,  is  determined  by  this 
standard. 

Like  numbers  ?  Unlike  ?  Commeneurable  ?  Incommensurable  ?  What  is  a 
given  number  ?    An  unknown  7    A  factor  ?    An  exact  divisor  ?    A  measure  ? 


82  PROPERTIES     OF     NUMBERS. 

105.  An  aliquot  part  of  a  number  is  a  factor  or  an 
ixact  divisor  of  that  number. 

Note. — The  terms  factors,  measures,  exact  divisors,  and  aliquot 
•parts,  are  different  names  of  the  same  thing,  and  are  often  iised  as 
synonymous.  Thus,  3  and  5  are  respectively  the  factors,  measures, 
exact  divisors,  and  aliquot  parts  of  15. 

106.  The  reciprocal  of  a  nurriber  is  i  divided  by 
that  number.    Thus,  the  reciprocal  of  4  is  1-7-4,  or  |. 


COMPLEMENT    OF    NUMBERS. 

107.  The  Complement  of  a  Nurriber  is  the  dif- 
ference between  the  number  and  a  unit  of  the  next  higher 
order.  Thus,  the  complement  of  7  is  3;  for  10  —  7  =  3; 
the  complement  of  85  is  15  ;  for  100  —  85  =  15. 

108.  To  find  the  Conipleinent  of  a  Number. 

Subtract  the  given  number  from  1  with  as  many  ciphers 
annexed,  as  there  are  iyitegral  figures  in  the  given  number. 

Or,  begin  at  the  left  hand,  and  subtract  each  figure  of  the 
given  number  from  9,  except  the  last  significant  figure  on 
tlhe  right,  which  must  be  tahen  from  10. 

Note. — The  second  method  is  based  upon  the  principle  that  when 
we  borrow,  the  next  upper  figure  must  be  considered  i  less  than  it  is. 
(Art.  37,  n.) 

Find  the  complement  of  the  following  numbers : 


I.  328. 

6.  6072. 

II.  56239. 

16. 

73245- 

2.  567. 

7.  8256. 

12.  64123. 

17- 

1234567- 

3.  604. 

8.  9061. 

13.  102345. 

18. 

2301206. 

4.  891. 

9.  13926. 

14.  261436. 

19. 

3021238. 

5.  4638. 

10.  23184. 

15.  40061. 

20. 

7830426. 

lo-;.  An  aliquot  part?  Note.  What  is  paid  as  to  the  use  of  these  fonr  terms? 
106.  The  reciprocal  of  a  number?  107.  The  complement?  108.  How  find  th« 
•omplement  of  a  number  ? 


PEOPERTiES   OF   :n^umbers.  83 


DIVISIBILITY    OF    NUMBERS. 

109.  One  number  is  said  to  be  divisible  by  anotlier, 
when  there  is  no  remainder.    The  division  is  then  com- 


When  there  is  a  remainder,  the  division  is  incomplete; 
and  the  dividend  is  said  to  be  indivisible  by  the  divisor. 

.  Notes. — i.  In  treating  of  the  divisibility  of  numbers,  the  term 
divisor  is  commonly  used  for  exact  dimsor. 

2.  Every  integral  number  is  divisible  by  the  unit  i,  and  by  itself. 
It  is  not  customary,  however,  to  consider  the  unit  i,  or  the  number 
itself,  as  &  factor.  (Art.  102.) 

110.  In  determining  the  divisibility  of  numbers  the 
ioWoyfmg pro2:)erties  ov facts  are  useful: 

Prop.  i.  Any  number  is  divisible  by  2,  which  ends  with  o,  2,  4, 
6,  or  8. 

2.  Any  number  is  divisible  by  3,  if  the  sum  of  its  digits  is  divisible 
by  3.  Thus,  147  the  sum  of  whose  digits  is  1+4  +  7=12,  is  divisible 
by  3. 

3.  Any  number  is  divisible  by  4,  if  its  ti'^o  right  hand  figures  are 
divisible  by  4;  as,  2256,  15368.  19384. 

4.  Any  number  is  divisible  by  5,  which  ends  with  o  or  5  ;  as, 
130,  675. 

5.  Any  even  number  is  divisible  by  6,  which  is  divisible  by  3. 
Thus,  1344-7-3=448  ;  and  1344-^-6=224. 

6.  Any  number  is  divisible  by  8,  if  its  three  right  hand  figures  are 
divisible  by  8  ;  as,  1840,  1688,  25320. 

7.  Any  number  is  divisible  by  10,  which  ends  with  o ;  by  100,  if  it 
ends  with  00 ;  by  1000,  if  it  ends  with  000,  etc. 

8.  Any  number  is  divisible  by  12  which  is  divisible  by  3  and  4. 

9.  Any  number  divided  by  9  will  leave  the  same  remainder  as  the 
warn  of  its  digits  divided  by  9.     Hence, 

10.  Any  number  is  divisible  by  9  if  the  sum  of  its  digits  is  divisible 
by  9.  Thus,  the  sum  of  the  digits  of  54378  is  5  +  4  +  3  +  7  +  8,  or  27. 
Now,  as  27  is  divisible  by  9,  we  may  infer  that  54378  is  divisible 
br9- 

10-,.  When  is  one  number  divisible  by  a.iotber  ?  When  indivisible  ?  no.  Whei 
Is  a  number  divisible  by  2 ?    By  3?    By  4?    By  5?    By  6?    By  8?    By  10? 


84  PKOPEKTIES     OF    NUMBERS. 

11.  As  9  is  a  multiple  of  3,  any  number  divisible  by  9,  is  also 
divisible  by  3. 

12.  Any  number  divided  by  11  will  leave  tbe  same  remainder  2ja 
the  sum  of  its  digits  in  the  even  places,  taken  from  the  sum  of  those 
in  the  odd  places,  counting  from  the  right,  the  latter  being  increased 
or  diminished  by  11  or  a  multiple  of  11.  Thus,  the  sum  of  the 
digits  in  the  even  places  of  314567,  viz.,  6  +  4  +  3,  ^^  '^3>  is  equal  to 
the  sum  of  the  digits  in  the  odd  places,  viz.,  7  +  5  +  1,  or  13 ;  there- 
fore, 314567  is  divisible  by  11.     Hence, 

13.  Any  number  is  divisible  by  11  when  the  sum  of  its  digits  in. 
the  even  places  is  equal  to  the  sum  of  those  in  the  odd  places,  or 
when  their  difference  is  divisible  by  1 1.  (For  a  demonstration  of  the 
properties  of  9  and  11,  see  Higher  Arithmetic.) 

14.  If  one  number  is  a  divisor  of  another,  the  former  is  also  a 
divisor  of  any  multiple  of  the  latter.  (Art.  103,  n.)  Thus,  2  is  a  divisor 
of  6  ;  it  is  also  a  divisor  of  the  product  of  3  times  6,  of  5  x  6,  and  of  any 
whole  number  of  times  6. 

15.  If  a  number  is  an  exact  divisor  of  each  of  two  numbers,  it  is 
also  an  exact  divisor  of  their  sum^  their  difference,  and  their  product. 
Thus,  3  is  a  divisor  of  9  and  15  respectively ;  it  is  also  a  divisor  ot 
9  +  15,  or  24;  of  15—9,  or  6;  of  15  X9,  or  135. 

16.  A  composite  number  is  divisible  by  each  of  its  prime  factors,  by 
the  product  of  any  two  or  more  of  them,  and  by  no  other  number. 
Thus,  the  prime  factors  of  30  are  2x3x5.  Now  30  is  divisible  by 
by  2,  3  and  5,  by  2  x  3,  by  2  x  5,  by  3  x  5,  and  by  2  x  3  x  5,  and  by  no 
other  number.    Hence, 

17.  The  least  divisor  of  a  composite  number,  is  a  prime  number. 

18.  An  odd  number  cannot  be  divided  by  an  even  number  <vithout 
a  remainder. 

19.  If  an  odd  number  divides  an  even  number,  it  will  also  divide 
half  of  it. 

20.  If  an  even  number  is  divisible  by  an  odd  number,  it  will  also 
be  divisible  by  double  that  number. 


If  a  number  is  divided  by  9,  to  what  is  the  remainder  equal  ?  When  ie  a 
Bumber  divisible  by  9?  If  a  number  is  divided  by  11,  to  what  is  the  remainder 
equal  ?  When  is  a  number  divisible  by  1 1  ?  If  a  number  is  a  divisor  of  another, 
what  is  true  of  it  in  regard  to  any  multiple  of  that  number  ?  If  a  number  is  a 
divisor  of  each  of  two  numbers,  what  is  true  of  it  in  regard  to  their  sum,  dif- 
ference, and  product  ?  By  what  is  a  composite  number  divisible  ?  Note.  What 
is  true  of  the  least  divisor  of  every  composite  number?  Is  an  odd  number 
dlrisible  by  an  even  ?  If  an  odd  divides  an  even  number,  what  is  true  of  half 
of  it  f  If  an  even  number  is  divisible  by  an  odd,  what  la  true  of  it  in  regard  to 
double  that  number  ? 


FACTORING. 

111.  Factors,  we  have  seen,  are  numbers  which  mul- 
tipUed  together,  produce  a  j9ro^wc^.     (Art.  42.)     Hence, 

112.  Factoring  a  number  is  finding  two  or  more  fac- 
tors, which  multipUed  togQi\\QY, produce  the  number.  Thus, 
the  factors  of  15  are  3  and  5  ;  for,  3  x  5  =  15.    (Art.  56,  n.) 

Note. — Every  number  is  divisible  by  itself  and  by  i ;  hence,  if 
multiplied  by  i,  the  product  will  be  the  number  itself.    (Art.  41.) 

But  it  cannot  properly  be  said  that  i  is  ^factor,  nor  that  a  nwriber 
is  a  factor  of  itself.     If  so,  all  numbers  are  composite.    (Art.  109,  n) 

113.  To  resolve  a  Composite  Number  into  tivo  Factors. 

Divide  the  numler  hy  any  exact  divisor  ;  the  divisor  and 
quotient  will  he  factors.     (Art.  62.) 

Note. — Every  composite  number  must  have  two  factors  at  least ; 
some  have  three  or  more  ;  and  others  may  be  resolved  into  several 
different  pairs  of  factors.    (Art.  loi,  Def.  6.) 

1.  What  are  the  two  factors  of  35  ?     Of  49  ?    Of  121  ? 

2.  Name  two  factors  of  45  ?     Of  56  ?     Of  7 2  ?    Of  io2  ? 

114.  To   find   the  Different  Pairs  of  factors   of  a 

Composite  Number. 

3.  What  are  the  different  pairs  of  factors  of  ^6  ? 

Analysis. — Dividing  36  by  2,  we  have  36-^2=18.  26-^-2  —  18 

Again,   36^3  =  12;  36h-4=:9;  and  36^6:116.     That  264-3=12 

is,  the  different  pairs  of  factors  of  36  are  2x18;  36-^-4=    9 

3x12;  4x9;  and  6x6.     Hence,  the  36  -f-  6  =:    6 

EuLE. — Divide  the  given  nurnber  continually  hy  each  of 
its  exact  divisors,  heginning  with  the  least,  until  the  quo^ 
iie?it  obtained  is  less  than  the  divisor  employed. 

T7ie  divisors  and  corresponding  quotie7its  will  he  the 
different  pairs  of  factors  required. 

III.  What  are  factors  ?  Is  i  a  factor?  Is  a  number  a  factor  of  itself ?  Why 
not  ?  What  is  meant  by  Factoring?  Note.  How  resolve  a  number  into  two  Iko 
tors  ?    113.  How  find  the  different  pairs  of  factors  ? 


86 


FACTOKIKG, 


Note.— The  division  is  stopped  as  soon  as  tlie  quotient  is  les»  than 
the  divisor;  for,  the  subsequent  factors  will  be  similar  to  those 
already  found. 


Find  the   different  pairs 
numbers : 

of   factors   of 

the 

following 

4.     20,               8.     sS, 
5-     27,               9,     45, 
6.     30,              10.     56, 

12.  96, 

13.  no, 

14.  144, 

16.  256, 

17.  475. 

18.  600, 

7-     32,              II.     75. 

15-     225, 

19.     1240. 

PRIME    FACTORS. 

115.  The  JPrime  Factors  of  a  number  are  the 
Prime  Numlers  which,  multiplied  together,  produce  the 
number. 

116.  Every  Composite  Wumher  can  be  resolved 
into  prime  factors.  For,  the  factors  of  a  composite  num- 
ber are  divisors  of  it,  and  these  divisors  are  either  prime 
or  composite.  If  prime,  they  accord  with  the  proposition. 
If  composite,  they  can  be  resolved  into  other  factors,  and 
so  on,  until  all  the  factors  are  prime.     (Art.  114.) 

117.  To  find  the  Frime  Factors  of  a  Composite  Number. 
Bx.  I.  What  are  the  prime  factors  of  210  ? 

Analysis.  —  Dividing    210  operation. 

by  any  prime  number  that        ist  divisor,  2       2 10,  given, 
will  exactly  divide  it,  as  2,        2d  "        3       105,  ist  quot. 

we  resolve  it  into  the  factors 
2  and  105.  Again,  dividing 
the  ist  quotient  105  by  any 
prime  number,  as  3,  we  re- 
solve it  into  the  factors  3  and 
35.  In  like  manner,  dividing  the  2d  and  3d  quotients  by  the  prime 
numbers  5  and  7,  the  4th  quotient  is  i.  The  divisors  2,  3,  5  and  7 
are  the  prime  factors  required.  For,  each  of  these  divisors  is  a 
prime  number,  and  the  division  is  continued  until  the  quotient  is  a 

115.  What  are  prime  factors?  ii6.  What  is  said  of  cempwiite  nmnbersf 
How  find  the  prime  factors  of  a  number  ? 


3d        "       5 

35.  2d 

4th       "       7 

7,3d 

I,  4th 

Hence,  210=2  x 

3x5x7. 

FACTOEIITG.  87 

unit ;   therefore,  fhe  several  divisors  must  "be  tlie  prime  factors 
required.     Hence,  the 

EuLE. — Divide  the  give7i  numher  hy  any  prime  number 
that  will  divide  it  without  a  remainder.  Again,  divide 
this  quotient  hy  a  prime  number,  and  so  on,  until  the  quo- 
tient obtained  is  i.  The  several  divisors  are  the  prime 
factors  required. 

Note. — As  the  least  divisor  of  every  number  is  prime,  beginners 
may  be  less  liable  to  mistakes  by  taking  for  the  divisor  the  smallest 
number  that  will  divide  the  several  dividends  without  a  remainder 

Find  the  prime  factors  of  the  following  numbers : 


2. 

72. 

7- 

184. 

12. 

1000. 

17- 

2348. 

3. 

96. 

8. 

215- 

13. 

1208. 

18. 

10376. 

4- 

121. 

9- 

320. 

14. 

1560. 

19. 

25600. 

5- 

^Z^' 

10. 

468. 

15. 

1776. 

20. 

64384. 

6. 

144. 

II. 

576. 

16. 

1868. 

21. 

98816, 

118.  To  find  the  Prime  Factors  common  to  two  or  more 
Numbers. 

22.  What  are  the  prime  factors  common  to  42,  168,  and 


210 


Analysis. — Dividing  the  given  numbers  by  the 

prime  factor  2,  the  quotients  are  21,  84,  and  105.  operation. 

Again,  dividing  these  quotients  by  the  prime  2)42,  168,  210 

factor  3,  the  quotients  are  7,  28,  and  35.     Finally,  ■5)2^      84   7o^ 

dividing  by  7,  the  quotients  are  i,  4,  and  5,  no  two        {-—^ -' 

of  which  can  be  divided  by  any  number  greater  1)    n     ^Q?     35 

than  I.     Therefore,  the  divisors  2,  3,  and  7  are  ij       4^       5 
the  common  prime  factors  required.     Hence,  the 

EuLE. — I.  Write  the  given  numbers  in  a  horizontal  line, 
and  divide  them  by  any  prime  number  ivhich  will  divide 
each  of  them  without  a  remainder. 

II.  Divide  the  quotients  thence  arising  in  the  same 
manner,  as  long  as  the  quotieyits  have  a  common  factor ; 
the  divisors  will  be  the  prime  factors  required. 

X18.  How  find  the  prime  factors  common  to  two  or  more  numbers? 


88  ^  CANCELLATIOir. 

Eind  the  prime  factors  common  to  the  following 
numbers : 

23.  12,  18,30.  29.  75,  90,  135,  150. 

24.  48,  66,  72.  30.   132,  144,  196,  240. 

25.  64,  108,  132.  31.  168,  256,  320,  500. 

26.  71,  113,  149.  32.  200,  325,  540,  625. 

27.  48,  72,  88,  120.  ;^;^.  316,  396,  484,  936. 

28.  60,  84,  108,  144.  34.  462,  786,  924,  858,  972. 

119.  To  find  the  Prime  Numbers  as  far  as  250. 

I.   TVrite  the  odd  nurnhers  in  a  series,  including  2. 

II.  After  3,  erase  all  that  are  divisible  hy  3  ;  after  5,  erase 
all  that  are  divisible  by  5  ;  after  7,  all  that  are  divisible  by  7  ; 
after  1 1  and  1 3,  all  that  are  divisible  by  them :  those  left 
are  primes. 

35.  Find  what  numbers  less  than  100  are  prime. 

S6,  Find  what  numbers  less  than  200  are  prime. 


CANCELLATION. 

120.  Cancellation  is  the  method  of  abbreviating  opera- 
tions by  rejecting  equal  factors  from  the  divisor  and  dividend. 

The  Slf/n  of  Cancellation  is  an  oblique  mark 
drawn  across  the  face  of  a  figure ;  as,  S,  5,  %  etc. 
Note. — The  term  cancellation  is  from  the  Latin  cancello,  to  erase. 

121.  Cancelling  a  Factor  of  a  number  divides 
the  number  by  that  factor.  For,  multiplpng  and  divid- 
ing a  number  by  the  same  factor  does  not  alter  its  value. 
(Art.  86,  a.)  Thus,  let  7  x  5  be  a  dividend.  Now  cancelling 
the  7,  divides  the  dividend  by  7,  and  leaves  5  for  the 
quotient ;  cancelling  the  5,  divides  it  by  5,  and  leaves  7 
for  the  quotient. 

I30.  What  ia  cancellation  ?    lai.  Effect  of  canoellJng  a  factor  ?    Wliy  ? 


CAKCELLATIOK.  89 

122.  To  divide  one  Composite  Number  by  another. 

1.  What  is  the  quotient  of  72  divided  by  18  ? 

Analysis. — The  dividend  72=9  x  4  x  2,  and  operation. 

tlie  divisor  18=9x2.     Cancelling  the  com-  72      ^  x  4  x  2. 

mon    factors  9  and    2,  we  have    4  for  the  ^^"^^2     ^^^ 
quotient. 

2.  Divide  the  product  of  45  x  17,  by  the  product  of  9  x  5^ 

Analysis. — In  this  example  the  product  of  the  two  9  ^5 

factors  composing  the  divisor,  viz. :   9x5,   equals  one  S  17 

factor  of  the  dividend.    We  therefore  cancel  these  equals  TZ7T 
at  once,  and  the  factor  17  is  the  quotient. 


17 


Bemark.—ln  this  operation  we  place  the  numbers  composing  the  divisor  on 
the  left  of  a  perpendicular  line,  and  those  composing  the  dividend  on  the  nght. 
The  result  is  the  same  as  if  the  divisor  were  placed  under  the  dividend  as  before. 
Each  method  has  its  advantages,  and  may  be  used  at  pleasure.    Hence,  the 

EuLE. — Caned  all  the  factors  common  to  the  divisor  and 
dividend,  and  divide  the  product  of  those  remaining  in  the 
dividend  ly  the  product  of  those  remaining  in  the  divisor, 
(Art.  ^6.) 

Notes. — i.  When  either  the  divisor,  dividend,  or  hotJi,  consist  of 
two  or  more  factors  connected  by  the  sign  x  ,  they  may  be  regarded 
as  composite  numbers,  and  the  common  factors  be  cancelled  before 
the  multiplication  is  performed. 

2.  This  rule  is  founded  upon  the  principle  that,  if  the  divisor  and 
dividend  are  both  divided  by  the  same  number,  the  quotient  is  not 
altered.  (Art.  86.)  Its  utility  is  most  apparent  in  those  operations 
which  involve  both  multiplication  and  division;  as  Fractions,  Pro- 
portion, etc. 

3.  When  the  factor  or  factors  cancelled  equal  the  number  itself, 
the  unit  i,  is  always  left ;  for,  dividing  a  number  by  itself,  the 
quotient  is  i.  When  the  i  stands  in  the  dividend,  it  must  be 
detained.    But  when  in  the  divisor,  it  is  disregarded.     (Art.  81.) 

3.  Divide  the  product  5  x  6  x  8  by  30  x  48. 

Solution. — Cancelling  the  common  5x6x8  5xlSxSi  i 
factors,  we  have  i  in  the  dividend  30  x  48  "=  S^x"^«r'  6 "^6 
and  6  in  the  divisor.     (Art.  112,  n.) 

.23.  Euie  for  cancelling  in  division  ?    Note.  On  what  is  the  rule  founded  ? 


90  ^  CA:N^CELLATIOi^. 

4.  Divide  24  x  6  by  12  x  7.         7.  Divide  18  x  15  by  5  x  6. 

5.  Divide  42  X  8  by  21  X  2.         8.  Divide  56  x  16  by  8  x  4. 

6.  18  X  3  X4by  12  X  6.  9.  28  x6  x  12  by  14  x  8. 

10.  Divide  21x5x6  by  7x3x2. 

11.  Divide  32x7x9  by  8x5x4. 

12.  Divide  27  X35  x  i4by9  x  7  x  21. 

13.  Divide  33x42x25  by  7  X5X11. 

14.  Divide  36  x  48  x  56  x  5  by  96  x  8  x  4  x  2. 

15.  Divide  63  X  24  X  33  X  2  by  9  X  12  X  II. 

16.  Divide  175  x  28  x  72  by  25  x  14  x  12. 

17.  Divide  220  x  60  x  48  x  69  by  13  x  no  x  12  x  8. 

18.  Divide  350  x6^x  144  by  50  x  72  x  24. 

19.  Divide  500  x  128  x  42  x  108  by  256  x  250  x  12. 

20.  How  many  barrels  of  flour,  at  $12  a  barrel,  must  bo 
given  for  40  tons  of  coal,  at  $9  per  ton  ? 

21.  How  many  tubs  of  butter  of  56  pounds  each,  at 
28  cents  a  pound,  must  be  given  for  8  pieces  of  shirting, 
containing  45  yards  each,  at  26  cents  a  yard  ? 

22.  Hovf  many  bags  of  coffee  42  pounds  each,  worth 
4  shillings  a  pound,  must  be  given  for  18  chests  of  tea, 
each  containing  72  pounds,  at  6  shillings  a  pound? 

23.  Bought  24  barrels  of  sugar,  each  containing  168 
pounds,  at  20  cents  a  pound,  and  paid  for  it  in  cheeses, 
each  weighing  28  pounds,  worth  16  cents  a  pound:  how 
many  cheeses  did  it  take  ? 

24.  A  grocer  sold  10  bogheads  of  molasses,  each  con- 
taining 6$  gallons,  at  7  shillings  per  gallon,  and  took  his 
pay  in  wheat,  worth  14  shillings  a  bushel:  how  much 
wheat  did  he  receive  ? 

25.  A  young  lady  being  asked  her  age,  replied:  If  you 
divide  the  product  of  64  into  14,  by  the  product  of  8  into 
4,  you  will  have  my  age :  what  was  her  age  ? 

26.  A  has  60  acres,  worth  $15  an  acre,  and  B  100  acres, 
worth  $75  an  acre:  B's  property  is  how  many  times  the 
value  of  A's  ? 


COMMON    DIVISORS. 

123.  A  Common  Divisor  is  any  number  that  will 
divide  two  or  more  numbers  without  a  remainder. 
124.  To  find  a  Coinnion  Divisor  of  two  or  more  Numbers, 

1.  Eequired  a  common  divisor  of  lo,  12,  and  14. 

Ai^ALYSis. — By  inspection,  we   perceive  tliat  ior=  10  =  2x5 

2x5;  12=2x6,  and  14=2x7.     The  factor  2,  is  com-  12  =  2x6 

mon  to  eacli  of  the  given  numbers,  and  is  therefore  14  =  2  X  7 

a  c;o/?iwi6>?i  divisor  of  them.     (Art.  123.)     Hence,  the  Ans.  2. 

Rule. — Resolve  each  of  the  given  numbers  into  two 
factors,  one  of  which  is  common  to  them  all. 

Find  common  divisors  of  the  following  numbers : 

2.  8,  16,  20,  and  24.  Ans.  2  and  4. 

3.  12,  15,  18,  and  30.  6.  21,  28,  35,  49,  d^. 

4.  36,  48,  96,  and  108.  7.  20,  30,  70,  and  100. 

5.  42,  54,  dd,  and  132.  8.  60,  75,  120,  and  240. 
9.  16,  24,  40,  64,  116,  120,  144,  168,  264,  1728. 

GREATEST    COMMON    DIVISOR. 

125.  The  Greatest  Common  Divisor  of  two  or 

more  numbers,  is  the  greatest  number  that  will  divide 
each  of  them  without  a  remainder.  Thus,  6  is  the  greatest 
common  divisor  of  18,  24,  and  30. 

Notes. — i.  A  common  divisor  of  two  or  more  numbers  is  the  same 
as  a  common  factor  of  those  numbers ;  and  the  greatest  common 
divisor  of  them  is  their  greatest  common  factor.     Hence, 

2.  If  any  two  numbers  have  not  a  common  factor,  they  cannot 
have  a  common  dinsor  greater  than  i.     (Art.  112,  n.) 

3.  A  common  divisor  is  often  called  a  common  measure,  and  the 
greatest  common  divisor,  the  greatest  common  measure. 

123.  A  common  divisor?  125.  Greatest  common  divisor?  Note.  A  common 
fllvisor  is  the  same  as  wliat  ?    Often  called  what  ? 


C0MM0  2^^     DIVISOES. 


FIRST    METHOD. 


126.  To  find  the  Greatest  Common  Divisor  by  continued 
Divisions. 

I.  What  is  the  greatest  common  divisor  of  28  and  40  ? 

Analysis. — If  we  divide  the  greater  number 
by  the  less,  the  quotient  is  i,  and  12  remainder.  operation. 

Next,  dividing  the  first  divisor  28,  by  the  first      28)40(1 
remainder  12,  the  quotient  is  2,  and  4  remainder.  28 

Again,  dividing  the  cecond  divisor  by  the  second 
remainder  4,  the  quotient  is  3,  and  o  rem.     The 


)28(: 


12    26(2 


last  divisor  4,  is  the  greatest  common  divisor.  —3 

Demonstration. — We  wish  to  prove  two  points :  4)^2(3 

1st,  That  4  is  a  common  divisor  of  the  given  num-  1 2 

bers.    2d,  That  it  is  their  greatest  common  divisor. 

First.  We  are  to  prove  that  4  is  a  common  divisor  of  28  and  40. 
By  the  last  division,  4  is  contained  in  12,  3  times.  Now,  as  4  is  a 
divisor  of  12  ;  it  is  also  a  divisor  of  i\ie  product  of  12  into  2,  or  24. 
(Art.  no,  Prop.  14.)  Next,  since  4  is  a  divisor  of  itself  and  24,  it 
must  be  a  divisor  of  the  sum  of  4  +  24,  or  28,  which  is  the  smaller 
number.  (Prop.  15.)  For  the  same  reason,  since  4  is  a  divisor  of  12 
and  28,  it  must  also  be  a  divisor  of  the  sum  of  12  +  28,  or  40,  which 
is  the  larger  number.     Hence,  4  is  a  common  divisor  of  28  and  40. 

Second.  We  are  now  to  prove  that  4  is  the  greatest  common  divisor 
of  28  and  40.  If  the  greatest  common  divisor  of  these  numbers  is 
not  4,  it  must  be  either  greater  or  less  than  4.  But  we  have  shown 
that  4  is  a  common  divisor  of  the  given  numbers;  therefore,  no 
number  less  than  4  can  be  the  greatest  common  divisor  of  them. 
The  assumed  number  must  therefore  be  greater  than  4.  By  supposi- 
tion, this  assumed  number  is  a  divisor  of  28  and  40;  hence,  it  must 
be  a  divisor  of  their  difference  40—28  or  12.  And  as  it  is  a  divisor  of 
12,  it  must  also  divide  the  product  of  12  into  2  or  24.  Again,  since 
the  assumed  number  is  a  divisor  of  28  and  24,  it  must  also  be  a  di- 
visor of  their  difference,  which  is  4  ;  that  is,  a  greater  number  will 
divide  a  less  without  a  remainder,  which  is  impossible.  (Art.  81, 
Prin.  2.)  Therefore,  4  must  be  the  greatest  common  divisor  of  28 
and  40.     Hence,  the 

126.  Explain  Ex.  i.  Prove  that  4  is  the  greatest  common  divisor  of  28  aad  40. 
Eule  ?  Note.  If  there  are  more  than  two  numbers,  how  proceed  ?  If  th<  Mst 
divisor  is  i,  then  what?  What  is  the  greatest  common  divisor  of  two  or  m«ra 
prime  numbers,  or  numbers  prime  to  each  other  ? 


COMMOiq^    DIVISORS.  93 

EuLE. — Divide  the  greater  nuniber  ly  the  less;  then 
divide  the  first  divisor  ly  the  first  remainder,  the  second 
divisor  by  the  second  remainder,  and  so  on,  until  nothing 
remains;  the  last  divisor  will  he  the  greatest  common 
divisor. 

Notes. — i.  If  there  are  more  tlian  two  mimbers,  begin  witli  the 
smaller,  and  find  the  greatest  common  divisor  of  two  of  them  ;  then 
of  this  divisor  and  a  third  number,  and  so  on,  until  all  the  numbers 
have  been  taken. 

2.  The  greatest  common  divisor  of  two  or  more  prime  numbers, 
or  numbers  prime  to  each  other,  is  i.     (Art.  112,  n.) 

3,  If  the  last  divisor  is  i,  the  numbers  are  prime,  or  prime  to  each 
other  ;  therefore  their  greatest  common  divisor  is  i.  Such  numbers 
are  said  to  be  incommensurable.    (Art.  loi,  Def  16.) 

2.  What  is  the  greatest  com,  divisor  of  48,  72,  and  108  ? 

3.  What  is  the  greatest  common  divisor  of  72  and  120? 

SECOND    METHOD. 

127.    To   find    the    greatest   Common  IHvisor    by  Prime 
Factors. 

4.  What  is  the  greatest  common  divisor  of  28  and  40  ? 
Analysis. — Resolving  the  given  numbers  into  operation 

factors   common  to  both,  we  have  28  =  2x2x7,        ^ 2x2  x*- 

and   40=2x2x10,     Now  the  product   of  these  ~~  ' 

factors,    viz.,  2  into  2,  gives  4  for  the  greatest      4°      2  X  2  X  10 
common  divisor,  tlio  same  as  by  the  first  method.      Ans.  2  X  2^4. 

5.  What  is  the  greatest  com.  div.  of  30,  45,  and  105  ? 

Analysts. — Setting  the  numbers  in  a  hori-  opekatiok. 

zontal  line,  we  divide  by  any  prime  nimaber,  as  3)30,  45,  105 

3,  that  will  divide  each  without  a  remainder,  cVio    iT      ^ 

and  set  the  quotients  under  the  corresponding 

numbers.     Again,  dividing  each  of  these  quo-  '     ^'        ' 

tients  by  the  prime  number  5,  the  new  quotients     3  -^  5^^^S>  ^^^« 
2,  3,  and  7,  are  prime,  and  have   no  common 
factor.     Therefore,  the  product  of  the  common  divisors  3  into  5,  or 
15,  is  the  giaatest  common  divisor.    (An.  no,  Prop.  16)    Hence,  the 

127.  What  is  the  rule  for  the  second  method?  Woie.  What  is  the  object  o/ 
placing  ths  numbers  in  a  horizontal  line  ? 


94  COMMOK     DIVISORS. 

Rule. — I.  Write  the  numbers  in  a  horizontal  line,  and 
divide  ly  any  prime  number  that  tvill  divide  each  without 
a  remainder,  setting  the  quotients  in  a  line  heloiv. 

II.  Divide  these  quotients  as  before,  and  thus  jiroceed  till 

•[10  number  can  be  found  that  loill  divide  all  the  quotients 

ivithout  a  remainder.     The  product  of  all  the  divisors  will 

be  the  greatest  common  divisor. 

Notes. — i.  This  rule  is  the  same  as  resolving  the  given  numbers 
into  prime  factors,  and  multiplying  together  all  that  are  common. 

2.  The  advantage  of  placing  the  numbers  in  a  horizontal  line,  is 
that  the  prime  factors  that  are  common,  may  be  seen  at  a  glance. 

3.  When  required  to  find  the  greatest  common  divisor  of  three  op 
more  numbers,  the  second  method  is  generally  more  expeditious,  and 
therefore  preferable. 

4.  When  the  given  numbers  have  only  one  common  factor,  that 
factor  is  their  greatest  common  divisor. 

5.  Two  or  more  numbers  may  have  several  common  divisors ;  but 
they  can  have  only  one  greatest  common  divisor. 

Find  the  greatest  common  divisor  of  the  following: 


6. 

67,  and  42. 

14. 

10,  28,  40,  64,  90,  32. 

7. 

135  and  105. 

15. 

12,  2>^,  60,  108,  132. 

8. 

24,  36,  72. 

16. 

16,  28,  64,  56,  160,  250. 

9- 

60,  75,  12. 

17- 

576  and  960. 

10. 

75,  125,  250. 

18. 

1225  and  592. 

II. 

42,  54,  60,  84. 

19. 

703  and  1369. 

12. 

72,  100,  168,  136. 

20. 

1492  and  1866. 

13. 

60,  84,  132,  108. 

21. 

2040  and  4080. 

22.  A  merchant  had  180  yards  of  silk,  and  234  yards 
of  poplin,  which  he  wished  to  cut  into  equal  dress  pat- 
terns, each  containing  the  greatest  possible  number  of 
yards :  how  many  yards  would  each  contain  ? 

23.  Two  lads  had  42  and  63  apples  respectively :  how 
many  can  they  put  in  a  pile,  that  the  piles  shall  be  equal, 
and  each  pile  have  the  greatest  possible  number  ? 

24.  A  man  had  farms  of  56,  72,  and  88  acres  respectively, 
which  he  fenced  into  the  largest  possible  fields  of  the  sam^ 
number  of  acres :  how  many  acres  did  he  put  in  each  ? 


MULTIPLES. 

128.  A  31ultiple  is  a  number  which  can  he  divided 
by  another  number,  without  a  remainder.  Thus,  12  is  a 
multiple  of  4. 

Remark. — The  Term  Multiple  is  also  used  in  the  sense 
of  product :  as  when  it  is  said,  "  if  one  number  is  a  di- 
visor of  another,  the  former  is  also  a  divisor  of  any  mul- 
tiple (product)  of  the  latter."  Thus,  3  is  a  divisor  of  6 ; 
it  is  also  a  divisor  of  7  times  6,  or  42. 

Note. — Multiple  is  from  the  Latin  multiplex,  having  many  folds, 
or  taken  many  times  ;  hence,  a  product. 

But  every  product  is  didsiUe  by  Ms  factors ;  hence,  the  term  came 
to  denote  a  dividend.  The  former  signification  is  derived  from  the 
formation  of  the  number ;  the  latter,  from  its  dimsihility. 

129.  A  Common  Multiple  is  any  number  that 
can  be  divided  by  two  or  more  numbers  without  a  re- 
mainder.   Thus,  1 8  is  a  common  multiple  of  2,  3,  6,  and  9. 

130.  A  common  multiple  of  two  or  more  numbers  may 
be  found  by  multiplying  them  together.  That  is,  the  pro- 
duct of  two  numbers,  or  any  entire  number  of  times  their 
product,  is  a  common  multifile  of  them.     (Art.  128,  ni) 

Notes. — i.  The  factors  or  divisors  of  a  multiple  are  sometimes 
called  sijh-muttiples. 

2.  A  number  may  have  an  unlimited  number  of  multiples.  For, 
according  to  the  second  definition,  every  number  is  a  multiple  of 
itself;  and  if  multiplied  by  2,  the  product  will  be  a  second  multiple  ; 
if  multiplied  by  3,  the  product  will  be  a  third  multiple ;  and  univer- 
sally, its  product  into  any  ichole  number  will  be  a  multiple  of  that 
number.     (Art.  128.) 

Find  a  common  multiple  of  the  following  numbers: 

1.  2,  3  and    5.         4.     2,    8  and  10.         7.  13,    7  and  22. 

2.  3,  7  and  II.         5.     7,    6  and  17.         8.  19,    2  and  40. 

3.  5,  7  and  13.         6.  11,  21  and  31.  9.   17,  10  and  34. 

128.  Meaning  of  the  term  multiple?  129.  What  is  a  common  multiple? 
130.  How  found?  Not6.  What  are  sub-multiples?  How  many  multiples  has  a 
number  ? 


96  MULTIPLES 


LEAST    COMMON    MULTIPLE. 

131.  The  Least  Common  Multiple  of  tvv^o  or 
more  numbers,  is  the  least  number  that  can  be  divided 
by  each  of  them  without  a  remainder.  Thus,  15  is  the 
least  common  multiple  of  3  and  5. 

132.  A  Multiple,  according  to  the  first  definition,  is  a 
coraijosite  number.  But  a  composite  number  contains  all 
the  prime  factors  of  each  of  the  numbers  which  produce 
it.     Hence,  we  derive  the  following  Principles : 

Prin.  I.  That  a  multiple  of  a  number  must  contain  all  the  prime 
factors  of  that  number. 

2,  A  common  multiple  of  two  or  more  numbers  must  contain  all 
the  prime  factors  of  each  of  the  given  numbers. 

3.  The  least  common  multiple  of  two  or  more  numbers  is  the  least 
number  which  contains  all  their  prime  factors,  each  factor  being 
tat  en  as  many  times  only,  as  it  occurs  in  either  of  the  given  numbers, 
and  no  more. 

133.    To  find  the  Least   Common   Multiple  of  two   or 
more  numbers.. 

I.  What  is  the  least  common  multiple  of  18,  21  and  66  ? 

ist  Method. — We  write  the  num-  ist  operation. 

bers  in  a  horizontal  line,  with  a  comma  2)18,  21    66 

between  them.     Since  2  is  a  prime  fac-  TT      ~    "~ 

tor  of  one  or  more  of  the  given  num-  -— ^-^ 

bers,  it  must  be  a  factor  of  the  least  3>       7>  ^  ^ 

common  multiple.  (Art.  132,  Prin.  i.)  2x3x3x7x11  =  1386 
We  therefore  divide  by  it,  setting  the 

quotients  and  undivided  numbers  in  a  line  below.  In  like  manner,  we 
divide  these  quotients  and  undivided  numbers  by  the  prime  number 
3,  and  set  the  results  in  another  line,  as  before.  Now,  as  the  numbers 
in  the  third  line  are  prime,  we  can  carry  the  division  no  further ; 
for,  they  have  no  common  divisor  greater  than  i,  (Art.  loi.) 
Hence,  the  divisors  2  and  3,  with  the  numbers  in  the  last  line, 
3.  7  and  II,  are  all  the  prime  factors  contained  in  the  given  numbers, 
and  each  is  taken  as  many  times  as  it  occurs  in  either  of  them. 

131.  What  is  the  leaBt  common  multipls?     132.  What  principles  arc  deriTcd? 


MULTIPLES.  97 

Therefore,  the  continued  product  of  these  factors,  2x3x3x7x11, 
or  1386,  is  the  least  common  multiple  required. 

2d  Method. — Resolving  each  num-  2d  operation. 

ber  into  its  prime  factors,  we  have  l8  =  2X3X    3 

18=2x3x3,  21  =  3x7,   and  66=2x3  21  =  3x7 

XII.     Now,  as  the  least  common  mul-  66  =  2x3x11 

tiple  must  contain  all  the  prime  fac- 

tors  of  the  given  number,  it  must  con-  2x3x11x7x3  =  13  86 
tain  those  of  66,  viz.,  2x3x11;    we 

therefore  retain  these  factors.  Again,  it  must  contain  the  prime 
factors  of  21,  which  are  3x7,  and  of  18,  which  are  2x3x3,  each 
being  taken  as  many  times  as  it  is  found  in  either  of  the  given 
numbers.  But  2  is  already  retained,  and  3  has  been  taken  once ;  we 
therefore  cancel  the  2  and  one  of  the  3s,  and  retain  the  other  3  and 
the  7.  The  continued  product  of  these  factors,  viz.,  2x3x11x7x3, 
is  1386,  the  same  as  before.     Hence,  the 

Rule. — I.  Write  the  numbers  in  a  horizontal  line,  and 
divide  ly  any  'prime  number  that  tvill  divide  tivo  or  more 
of  them  without  a  remainder,  placing  the  quotients  and 
numbers  U7idivided  in  a  line  below. 

II.  Divide  this  line  as  before,  and  thus  proceed  till  no 
two  numbers  are  divisible  by  any  number  greater  than  i. 
Tlie  continued  product  of  the  divisors  and  numbers  in  the 
last  line  will  be  the  answer. 

Or,  resolve  the  mimbers  into  their  prime  factors  ;  mid- 
iiply  these  factors  together,  talcing  each  the  greatest  number 
of  times  it  occurs  in  either  of  the  given  numbers.  The 
product  will  be  the  ansiuer. 

Notes. — i.  These  two  methods  are  based  upon  the  same  principle, 
viz. :  that  the  least  common  multiple  of  two  or  more  numbers  is  the 
least  number  which  contains  all  their  prime  factors,  each  factor 
being  taken  as  many  times  only,  as  it  occurs  in  either  of  the  given 
numbers.     (Art.  132,  Prin.  3.) 

2.  The  reason  we  employ  prime  numbers  as  divisors  is  because  the 
given  numbers  are  to  be  resolved  into  prime  factors,  or  factors  ^rm6 

133.  How  find  the  least  common  multiple  ?  Note.  Upon  what  principle  are  these 
two  methods  hased  ?  Why  employ  prime  numbers  for  divisors  in  the  first 
method  ?  Why  write  the  numbers  in  a  horizontal  line  ?  What  is  the  second 
method  ?  Advantage  of  it  ?  How  may  the  operation  be  shortened  ?  When  the 
given  numbers  are  prime,  or  prime  to  each  other,  what  is  to  be  done  ? 


98  MULTIPLES. 

to  each  other.  If  the  divisors  were  composite  numbers,  they  would 
be  liable  to  contaiu  factors  common  to  some  of  the  quotients,  or 
numbers  in  the  last  line ;  and  if  so,  their  continued  product  would 
not  be  tlie  least  common  multiple. 

3.  The  object  of  placing  the  numbers  in  a  horizontal  line,  is  to 
resolve  all  the  numbers  into  prime  factors  at  the  same  time. 

4.  The  chief  advantage  of  the  second  method  lies  in  the  distinct- 
ness with  which  the  prime  factors  are  presented.  The  former  is  the 
more  expeditious  and  less  liable  to  mistakes. 

5.  The  operation  may  often  be  shortened  by  cancelling  any  num- 
ber which  is  a  factor  of  another  number  in  the  same  line.  (Ex.  2.) 
When  the  given  numbers  are  prime,  or  prime  to  each  other,  they 
have  no  common  factors  to  be  rejected ;  consequently,  their  con- 
tinued product  will  be  the  least  common  multiple.  Thus,  the  least 
common  multiple  of  3  and  5  is  15  ;  of  4  and  9  is  36. 

2.  Find  the  least  common  multiple  of  5,  6,  9,  10,  21. 

Analysis. — In  the  first  line  5  2)5,  6,  9,  10,  21 

is  a  factor  of  10,  and  is  therefore  ^w  I     ~ 

cancelled  ;  in  the  second,  3  is  a  — ^—^ — 

lactor  of  9,  and  is  also  cancelled.  3?     5?     7 

The  product  of  2X3X3X5X  7=630,  the  answer  required. 

Find  the  least  common  multiple  of  the  following : 

10.  2>^,  48,  72,  96. 
42,  6^^  84,  108. 
120,  144,  168,  216. 
96,  108,  60,  204. 
126,  154,  280,  560. 
i44>  256,  72,300. 
250,  500,  1000. 

17.  Investing  the  same  amount  in  each,  what  is  the 
smallest  sum  with  which  I  can  buy  a  whole  number  of 
pears  at  4  cents,  lemons  at  6,  and  oranges  at  10  cents  ? 

18.  A  can  hoe  16  rows  of  corn  in  a  day,  B  18,  C  20,  and 
D  24  rows :  what  is  the  smallest  number  of  rows  that  will 
keep  each  employed  an  exact  number  of  days  ? 

19.  A  grocer  has  a  4  pound,  5  pound,  6  pound,  and  a 
T2  pound  weight:  what  is  the  smallest  tub  of  butter  that 
can  be  weighed  by  each  without  a  remainder  ? 


3. 

8,  12,  16,  24. 

10. 

4. 

14,  28,  21,  42. 

II. 

5 

36,  24,  48,  60. 

12. 

6. 

25?  40,  75.  loo- 

13. 

7- 

16,  24,  ^2,  40. 

14. 

8. 

22,  2>3,  SS,  66. 

15. 

9- 

30,  40,  60,  80. 

16. 

FRACTIONS. 

134.  A  Fraction  is  one  or  more  of  the  equal  parts 
into  which  a  unit  is  divided. 

The  numher  of  these  parts  indicates  their  name.  Thus, 
when  a  unit  is  divided  into  tiuo  equal  parts,  the  parts  are 
CdXl^di  halves ;  when  into  three,  thej  are  called  thirds;  when 
into  four,  fourths,  etc.     Hence, 

135.  A  half  is  one  of  the  iivo  equal  parts  of  a  unit ;  a 
third  is  one  of  the  three  equal  parts  of  a  unit;  a  fourth, 
one  of  the  four  equal  parts,  etc. 

Note. — The  term  fraction  is  from  the  Latin  frango,  to  Ireak. 
Hence,  Fractions  are  often  called  Broken  Numbers. 

136.  The  value  of  these  equal  parts  depends, 
First.  Upon  the  magnitude  of  the  unit  divided. 
Second.  The  numher  of  parts  into  wiiich  it  is  divided. 

Illustration. — ist.  If  a  large  and  &  small  apple  are  each  divide  J 
into  two,  three,  four,  etc.,  equal  parts,  it  is  plain  that  the  parts  of  the 
former  will  be  larger  than  the  corresponding  parts  of  the  latter. 

2d.  If  one  of  two  equal  apples  is  divided  into  two  equal  parts,  and 
the  other  mto  four,  the  parts  of  the  first  will  be  twice  as  large  as 
those  of  the  second ;  if  one  is  divided  into  two  equal  parts,  the  other 
into  six,  one  part  of  the  first  will  be  equal  to  three  of  the  second,  etc 
Hence, 

Note. — A  Jialf  is  twice  as  large  as  ?i  fourth,  three  times  as  large  as 
a  sixth,  four  tim.es  as  large  as  an  eighth,  etc. ;  and  generally, 

The  greater  the  number  of  equal  parts  into  which  the  unit  is  di- 
vided, the  less  wi!l  be  the  value  of  each  part.     Conversely, 

The  less  the  number  of  equal  parts,  the  greater  will  be  the  value 
of  each  part. 


134.  What  is  a  fraction  ?  What  does  the  number  of  parts  indicate  ?  135,  What 
is  a  half  ?  A  third ?  A  fourth ?  A  tenth  ?  A  hundredth  ?  136.  Upon  what  does 
the  size  of  these  parts  depend  ?  99 


100  FBACTIONS. 

137.  Fractions  are  divided  into  two  classes,  common  and 
decimal. 

A  Common  Fraction  is  one  in  which  the  unit  is 
divided  into  any  number  of  equal  parts. 

138.  Common  fractions  are  expressed  by  figures  writ- 
ten above  and  below  a  line,  called  the  numerator  and 
denominator;  as  -|,  f,  ^. 

139.  The  Denominator  is  written  lelow  the  Hue, 
and  shows  into  lioio  many  equal  parts  the  unit  is  divided. 
It  is  so  called  because  it  names  the  parts ;  as  halves,  thirds, 
fourths,  tenths,  etc. 

The  Numerator  is  written  above  the  line,  and 
shows  Uoiv  many  parts  are  expressed  by  the  fraction.  It 
is  so  called  because  it  numlers  the  parts  taken.  Thus,  in 
the  fraction  f,  four  is  the  denominator,  and  show^s  that 
the  unit  is  divided  vnio  four  equal  parts;  tliree  is  the  nu- 
merator, and  shows  that  tliree  of  the  parts  are  taken. 

140.  The  Terms  of  a  fraction  are  the  numerator  and 
denominator. 

Note. — Fractions  primarily  arise  from  dividin^^  a  single  unit  into 
equal  parts  They  are  also  used  to  express  a  part  of  a  collection  of 
units,  and  apart  of  a.  fraction  itself;  as  i  half  of  6  pears,  i  third  of 
3  fourths,  etc.  But  that  from  which  they  arise,  is  always  regarded 
as  a  iDhole,  and  is  called  the  Unit  or  Base  of  the  fraction. 

141.  Common  fractions  are  usually  divided  into  proper, 
improper,  simple,  compound,  complex,  and  mixed  numbers. 

1.  A  Proper  Fraction  is  one  whose  numerator  is 
less  than  the  denominator;  as  ^,  f,  |. 

2.  An  Imjyroj^er  Fraction  is  one  w^hose  numerator 
equals  or  exceeds  the  denominator;   as  f,  f. 

3.  A   Sitnple    Fraction   is    one    having  but  one 

137.  Into  how  many  elapses  are  fractions  divided?  A  common  fraction? 
138.  How  expressed  ?  139.  What  does  the  denominator  show  ?  Why  so  called? 
The  numerator  ?    Why  so  called  ?     140.  What  are  the  terms  of  a  fraction  ? 


FRACTIOI^S.  10] 

numerator  and  one  denominator,  eacli'of  whdcli  is  d^'xvkole 
number,  and  may  be  proper  or  improper ;    as  f ,  f . 

4.  A  Compound  Fraction  is  a  fraction  of  a  frac- 
tion ;  as  J  of  f . 

5.  A  Co^nple'x  Fraction  is  one  which  has  a  frac- 

I    2i 
tional  numerator,  and  an  integral  denominator;  as,  -,  — . 

4  5 
Remark. — Those  abnormal  expressions,  haring  fractional  denoyn- 
inators,  commonly  called  complex  fractions,  do  not  strictly  come  under 
the  definition  of  a  fraction.  For,  a  fraction  is  one  or  more  of  the 
equal  parts  into  which  a  unit  is  divided.  But  it  cannot  properly  be 
said,  that  a  unit  is  divided  into  Sjtlis,  45dths,  etc. ;  that  is,  into  3^ 
equal  parts,  4I  equal  parts,  etc.  They  are  expressions  denoting  divi- 
sion, having  o,  fractional  divisor,  and  should  be  treated  as  such. 

6.  A  llixed  dumber  is  a  whole  number  and  a  frac- 
tion expressed  together ;  as,  5 1,  34^^. 

Note. — The  primary  idea  of  a  fraction  is  a  part  of  a  unit.  Hence, 
a  fraction  of  less  value  than  a  unit,  is  called  a  proper  fraction. 

All  other  fractions  are  called  improper,  because,  being  equal  to  or 
greater  than  a  unit,  they  cannot  be  said  to  be  a  pa7't  of  a  unit. 

Express  the  following  fractions  by  figures : 

1.  Two  fifths.  6.  Thirteen  twenty-firsts. 

2.  Four  sevenths.  7.  Fifteen  ninths. 

3.  Three  eighths.  8.  Twenty-three  tenths. 

4.  Five  twelfths.  9.  Thirty-one  forty-fifths. 

5.  Eleven  fifteenths.      10.  Sixty-nine  hundredths. 

11.  112  two  hundred  and  fourths. 

12.  256  five  hundred  and  twenty-seconds. 

13.  Explain  the  fraction  -J. 

Analysis. — \  denotes  i  of  i ;  that  is,  one  such  part  as  is  obtained 
by  dividing  a  unit  into  4  equal  parts.  The  denominator  names  the 
parts,  and  the  numerator  numbers  the  pirts  taken.  It  is  common 
because  the  unit  is  divided  into  any  number  of  parts  taken  at  ran- 
dom ;  proper,  because  its  value  is  less  than  i  ;  and  simple,  because  it 
has  but  one  numerator  and  one  denominator,  each  of  which  is  a 
whole  number. 


141 .  Into  what  are  common  fractions  divided  ?  A  proper  fraction  ?   Improper  ? 
Simple  ?    Compound  ?    Complex  ?    A  mixed  number  ? 


15- 

*. 

20. 

¥• 

1 6. 

ioff. 

21. 

fl- 

17- 

«• 

22. 

If. 

1 8. 

fofif. 

23- 

If. 

19. 

4i 

8* 

24. 

10 

102  FRACTIOI^S. 

14/  Explain  the  fraction  f. 

Analysis. — }  denotes  f  of  i,  or  ^  of  3.  For,  if  3  equal  lines  are 
each  divided  into  4  equal  parts,  3  of  these  parts  will  be  equal  to  J  of 
I  line,  or  ^  of  the  3  lines.     It  is  common,  etc.     (Ex.  13.) 

Read  and  explain  the  following  fractions : 

25.  1 6 J. 

26.  28fL 

27.  J  off  of  |. 

28.  f  of  f  of  2j. 

29.  — ^. 

Note. — The  21st  and  22d  should  be  read,  "  68  twenty-Jlrsts,*' 
"  85  thhty-seconds,"  and  not  68  twenty-ones,  85  thirty-twos. 

142.  Fractions,  we  haye  seen,  arise  from  division,  the 
mcmerator  being  the  dividend,  and  the  denorninator  the 
divisor.     (Art.  64,  ?2.)     Hence, 

The  value  of  a  fraction  is  the  quotient  of  the  numerator 
divided  by  the  denominator.  Thus,  the  value  of  i  fourth 
is  I -i- 4,  or  J;  of  6  halves  is  6-^-2,  or  3;  of  3  thirds  is 
3-^3,  or  I. 

143.  To  find  a  Fractional  Part  of  a  number. 

I.  What  is  I  half  of  12  dimes? 

ANx\lysis. — If  12  dimes  are  divided  into  2  equal  parts,  i  of  these 
parts  will  contain  6  dimes.     Therefore,  2  of  12  dimes  is  6  dimes. 

Note. — The  solution  of  this  and  similar  examples  is  an  illustra- 
tion of  the  second  object  or  office  of  Division.     (Art.  63,  &.) 

4.  What  is  J  of  36  ?     -I-  of  45  ?     |  of  60  ?     |  of  6^  ? 

5.  What  is  I  of  48  ?    ^  of  63?    -^  of  190?    ^^  of  132? 

6.  What  is  J  of  12  dimes? 

Analysis. — 2  thirds  are  2  times  as  much  as  3.  But  i  third  of  12 
is  4.    Therefore,  §  of  12  dimes  are  2  times  4,  or  8  dimes.    Hence,  the 

142.  From  what  do  fractions  arise  ?  Which  part  is  the  dividend  ?  Which  the 
divisor  ?    What  is  the  value  of  a  fraction  ? 


FRACTIOJ^^S.  103 

Rule. — Divide  the  give^i  number  hy  the  denominator,  and 
multiply  the  quotient  by  the  numerator 

To  find  J,  divide  the  number  bj  2. 
To  find  J,  divide  the  number  by  3. 
To  find  J,  divide  by  3,  and  multiply  by  2,  etc. 

7.  What  is  J  of  56  ?  Ans.  42. 

8.  What  is  f  of  30  apples  ?     Of  45  ?     Of  60  ? 

9.  What  is  f  of  42  ?     1^  of  56  ?     f  of  72  ? 

10.  What  is  I  of  45  ?    A  of  50?    ^^  of  66  ?    ^'^  of  108? 


GENERAL   PRINCIPLES   OF   FRACTIONS. 

144.  Since  the  numerator  and  denominator  have  the 
mme  relation  to  each  other  as  the  dividend  and  divisor,  it 
follows  that  the  general  principles  established  in  division 
are  applicable  to  fractions.     (Art.  81.)     That  is, 

1.  If  the  numerator  is  equal  to  the  denominator,  the 
value  of  the  fraction  is  i. 

2.  If  the  denominator  is  greater  than  the  numerator,  the 
value  is  less  than  i. 

3.  If  the  denominator  is  less  than  the  numerator,  the 
value  is  greater  than  i. 

4.  If  the  denominator  is  i,  the  value  is  the  numerator. 

5.  Multiplying  the  numerator,  multiplies  the  fraction. 

6.  Dividing  the  numerator,  divides  the  fraction. 

7.  Multiplying  the  denominator,  divides  the  fraction. 

8.  Dividing  the  denominator,  multiplies  the  fraction. 

9.  Multiplying  or  dividing  both  the  numerator  a7id  de- 
nominator by  the  same  number  does  not  alter  the  value  of 
the  fraction. 


144.  If  the  numerator  is  equal  to  the  denominator,  what  is  the  value  of  the 
fraction?  If  the  denominator  is  the  jjreater,  what?  If  less,  what?  If  the  de- 
nominator is  r,  what  ?  What  is  the  effect  of  multiplying  the  numerator  ?  Divid- 
ing the  numerator  ?    Multiplying  the  denominator  ?    Dividing  the  denominator  ? 


104  FEACTIOIfS. 


REDUCTION    OF    FRACTIONS. 

145.  deduction  of  fractions  is  changing  their  terms, 
Tvdthout  altering  the  value  of  the  fractions.     (Art.  142.) 

CASE   I. 

146.  To  Reduce  a  Fraction  to  its  Lowest  Terms. 

Der — The  Lowest  Terms  of  a  fraction  are  the 
smallest  numbers  in  which  its  numerator  and  denominator 
can  be  expressed.     (Art.  10 1,  Def.  7.) 

Ex,  I.  Eednce  fl  to  its  lowest  terms. 

Analysis. — Dividing  both  terms  of  a  fraction  ^st  method. 

by  the  same  number  does  not  alter  its  value.        .24 12 

(Art.  144,  Prin.  9.)      Hence,  we  divide  the  given  ^^^       16 

terms  by  2,  and  the  terms  of  the  new  fraction  j  2        -> 

by  4  ;  the  result  is  |.    Bat  the  terms  of  the  frac-  4)77  —  ~     A7IS. 
tion  I  are  prime  to  each  other ;  therefore,  they 


16       4 


Or,  if  we  divide  both  terms  by  their  greatest  2d  method. 

common  divisor,  which  is  8,  we  shall  obtain  the     qx24     3 
same  result.    (Art.  126.)    Hence,  the  ^32  ~4^^^* 

Rule. — Divide  tli^  numerator  and  denominator  C07i- 
tinualltf  hy  any  numher  that  will  divide  loth  without  a  re- 
mainder, until  no  numher  greater  than  i  will  divide  them. 

Or,  divide  hoih  terms  of  the  fraction  hy  their  greatest 
common  divisor.    (Art.  126.) 

Notes. — i.  It  follows,  conversely,  that  a  fraction  is  reduced  to 
higher  terms,  by  multiplying  the  numerator  and  denominator  by  a 
common  multiplier.    Thus,  §=i^,  both  terms  being  multiplied  by  8. 

2.  These  rules  depend  upon  the  principle,  that  dividing  both  terms 
of  a  fraction  by  the  same  number  does  not  alter  its  value.  When  the 
terms  are  large,  the  second  method  is  preferable. 

145.  What  is  Reduction  of  Fractions?  146.  What  are  the  lowest  terms  of  a 
fraction  ?    How  reduce  a  fraction  to  its  lowest  terms  t 


KEDUCTIOK     or     FEACTIOKS. 


105 


Reduce  the  following  fractions  to  their  lowest  terms 


2.  if- 

9- 

a- 

16.  ilf. 

23- 

-fo%- 

3-U- 

10. 

m- 

17.  ih 

24. 

m-i- 

4.  H- 

II. 

m- 

18.  in- 

25- 

-m- 

5-  if- 

12. 

Hi- 

19-  i¥oV 

26. 

#^A- 

6.U- 

13- 

m- 

20.  m- 

27. 

mf- 

1-U- 

14. 

t¥^- 

21-  t¥A. 

28. 

ilU- 

s.ki- 

15- 

«• 

CASE 

22.  i¥^V 
II. 

29. 

AV 

7.    To 

re#uce   an   Im 

proper 

Fraction    to 

a    Whole   or 

M'lQced  Number, 

I.  Reduce  ^^-  to  a  whole  or  mixed  number. 

Analysis. — Since  4  fourths  make  a  unit  or  i,  45 
fourths  will  make  as  many  units  as  4  is  contained 
times  in  45,  which  is  ii;^.     Hence,  the 


4)45 


iiiAns. 


Rule. — Divide  the  numerator  hy  tlie  denominator. 

Notes. — i.  This  rule,  in  effect,  divides  both  terms  of  the  fraction 
by  the  same  number ;  for,  removing  the  denominator  cancels  it,  and 
cancelling  the  denominator  divides  it  by  itself.     (Art.  121.) 

2.  If  fractions  occur  in  the  answer,  they  should  be  reduced  to  the 
loijcest  terms. 


Reduce  the  following  to  whole  or  mixed  numbers : 


.  iji. 

8. 

^F. 

14. 

hS'^- 

20. 

mi^- 

. -^. 

9- 

w. 

15- 

¥t¥- 

21. 

ift&r- 

^.  ^i^. 

10. 

%^-. 

16. 

10000. 

22. 

^S:V/^. 

.  W. 

II. 

nf. 

17. 

¥A\^- 

23- 

4fVW/. 

.  i^. 

12. 

w. 

18. 

mi^. 

■   24. 

^t"-.^- 

.  ^^^ 

13- 

fM. 

19. 

mi^- 

25- 

'fih'^- 

26.  Ii\  ^|4?^'*-  of  a  pound,  how  many  pounds  ? 

27.  In  ^^^^-i^  of  a  dollar,  how  many  dollars  ? 

28.  In  ^^j|g^"  of  9^  y^ar,  bow  many  years? 


How  reduce  an  imp.'"oper  fraction  to  a  wLole  or  mixed  number  f 


106  KEDUCTIOiT     OF     FRACTIOKS. 

CASE    III. 
148.  To  reduce  a  Mixed  Number  to  an  I^nproper  FracUoru 

I.  Reduce  8f  to  an  improper  fraction. 

Analysis. — Since  there  are  7  sevenths  in  a  unit,  there  03 
must  be  7  times  as  many  sevenths  in  a  number  as  there  ,.  "^ 
are  units,  and  7  times  8  are  56  and  3  sevenths  make  ^7^.  j~ 
Therefore,  8^=^7^.  ^'  ^^5- 

Or  thus :  Since  i=f,  8  =  8  times  ^  or  ^7^,  and  f  make  ^7^.  In  the 
operation  we  multiply  the  integer  by  the  given  denominator,  and  to 
the  product  add  the  numerator.     Hence,  the 

EuLE. — Multiply  the  whole  number  ly  the  given  denomi- 
nator;  to  the  product  add  the  numerator,  and  place  the 
mm  over  the  denominator. 

Eemark. — A  wJiole  number  may  be  reduced  to  an  improper 
fraction  by  making  i  its  denominator.     Thus,  4=^.     (Art.  81.) 

Reduce  the  following  to  improper  fractions : 


2-  i5f 

6. 

HSii- 

10. 

i573f 

14.  478^- 

3.  i8f. 

7- 

l8^^. 

II. 

2564. 

15.  57AV 

4-  35i 

8. 

295:^V- 

12. 

3640!. 

16.  8|«. 

5.  SiU- 

9- 

806^?^. 

13- 

8624A. 

17.  9^'A. 

18.  In  263y\  pounds  how  many  sixteenths  ? 

19.  Change  641^^  mile  to  fortieths  of  a  mile. 

CASE    IV. 
149.  To  reduce  a  Compound  Fraction  to  a  Simple  one, 

I.  Reduce  J  of  f  to  a  simple  fraction. 

Analysis. — .3  fourths  of  |  =  3  times  i  fourth  of  f.  ist  method. 

Now  I  fourth  of  |  is  -,% ;  for  multiplying  the  de-       3      2 6 

nominator  divides  the  fraction;   and  3   fourths   of  4     7~~72 
1  =  3  times  -,%  or  ,^^  ;  for  multiplying  the  numerator 
multiplies  the  fraction.     (Art.  144.)    Reduced  to  its  J^«  — -zn- 
lowest  terms  1^2=^.  '12      2 

148.  How  reduce  a  mixed  number  to  an  improper  fraction?    Eem.  A  whole 
number  to  an  improper  fraction  ? 


EEDUCTIOK     OF     FRACTIOKS.  10? 

Or,    since    numerators    are    dividends    and  3d  method. 

A  /nominators    divisors,   the    factors   2   and   3,  i,  S     2,      1 

which  are  common  to  both,  may  be  cancelled.  2   4  ^  ^^^     Ans. 

(Art.  144,  Prin.  9.)    The  result  is  |.    Hence,  the  ' 

Rule. — Cancel  the  common  factors,  and  2:)lace  the  product 
of  the  factors  remaining  in  the  numerators  over  the  product 
of  those  remaining  in  the  denominators. 

Notes. — i.  Whole  and  mixed  numbers  must  be  reduced  to  irriproper 
fractions,  before  multiplying  the  terms,  or  cancelling  the  factors. 

2.  Cancelling  the  common  factors  reduces  the  result  to  the  lowest 
terms,  and  therefore  should  be  employed,  whenever  practicable. 

3.  The  numerators  being  dividends,  may  be  placed  on  the  right 
of  a  perpendicular  line,  and  the  denominators  on  the  left,  if  pre- 
ferred.   (Art.  122,  Rem.) 

4.  The  reason  of  the  rule  is  this:  multiplying  the  numerator  of 
one  fraction  by  the  numerator  of  another,  multiplies  the  value  of  the 
former  fraction  by  as  many  units  as  are  contained  in  the  numerator 
of  the  latter ;  consequently  the  result  is  as  many  times  too  large  as 
there  are  units  in  the  denominator  of  the  latter.  (Art.  144,  Prin.  5.) 
This  error  is  corrected  by  multiplying  the  two  denominators  to- 
gether,    (Art.  144,  Prin,  7.) 

2.  Reduce  f  of  J  of  ^V  of  2  J  to  a  simple  fraction.   Ans.  j|. 

Reduce  the  following  to  simple  fractions : 

3. 1  of  ^\.  9. 1  of  2V  of  60.  15.  ^3  of  f  I  of  iJ. 

4.  f  of  f  10.  ^  of  li  of  if.  16.  J  of  f  of  I  of  f 

5.  f  of  M-.  1 1 .  H  of  li  of  f  J.  17.  ii  of  IJ  of  65  J. 
5.  f  of  if-  12.  If  of -I  of  If.  18.  fi  of  If  of  84i 

7.  f  of  if.  13.  M  of  /y  of  45.  19.  I  of  ^  of  ig. 

8.  f  of  tV-  14.  f  of  I  of  f  J.  20.  f  of  i  of  163  J. 

21.  Reduce  ^f  of  f^  of  f-J  of  9!  to  a  simple  frac- 
tion. 

22.  To  what  is  I  of  fj  of  3^  bushels  equal? 
2T,.  To  what  is  f  of  4 J  of  ^  of  a  yard  equal  ? 


-49,  How  reduce  a  compound  fraction  to  a  simple  one  ?    Note.  What  must  bo 
<Ione  with  whole  and  mixed  numbers  ?    What  is  the  advantage  of  cancelling  ? 


108  REDUCTION     OF     FRACTIONS. 

CASE     V. 
150.  To  reduce  a  Fraction  to  any  required  Denominatcr, 

1.  Reduce  f  to  twenty-fourths. 

1ST  Analysis. — 8  is  contained  in  24,  3  times ;  tliere-  24-7-  8  =  t 
fore,  multiplying  both  terms  by  3,  the  fraction  be-  6  X  ^  18 
comes  il,  and  its  value  is  not  altered.     (Art.  144.)  o  ^  ~  ^^  ~ 

Rem. — If  required  to  reduce  |  to  fourths,  we  should  divide  both 
terms  by  2,  and  it  becomes  f .     (Art.  144.) 

2D  Analysis. — Multiplying    both    terms  of   f  6x24=144 

by  24,  the    required    denominator,   we   have  |^|-.     0  ^  ^. Z~Z 

(Art.  144,  Prin.  9.)    Again,  dividing  both  terms  of  |^f  ,  ^ ^ 

by  8,  the  given  denominator,  we  have  I4,  the  same    -—  *  

as  before.     Hence,  the  1 92  -^-  8  =  24 

Rule. — IfuUijjlt/  or  divide  loth  terms  of  the  fraction 
hy  such  a  numher  as  will  make  the  given  de7iominator 
equal  to  the  required  denominator. 

Or,  Multiply  loth  terms  of  the  given  fraction  ly  the 
required  denominator;  then  divide  loth  terms  of  thv  re- 
sult ly  the  given  denominator.     (Art.  144,  Prin.  9.) 

Notes. — i.  The  multiplier  required  by  the  ist  method  is  found 
by  dividing  the  proposed  denominator  by  the  given  denominator. 

2.  When  the  required  denominator  is  neither  a  midtiple,  nor  an 
exact  divisor  of  the  given  denominator,  the  result  will  be  a  complex 

fraction.     Thus,  ^  reduced  to  tenths:^— ;  f  reduced  to  sevenths=— - 

Change  the  following  to  the  denominator  indicated : 

2.  ^  to  fifty-seconds.  7.  ||-  to  246ths. 

3.  tV  to  sixtieths.  8.  f  J  to  2  88ths. 

4.  yV  to  eighty-eighths.  9.  |-J  to  36oths. 

5.  iJto  i44ths.  10.  -^  to  loooths. 

6.  Jl  to  204ths.  II.  ^Vcnr  to  looooths. 

ji 

12.  Reduce  4- to  fourths.  Ans.  — . 

^  4 

1 3.  Reduce  fV  to  sixths.    15.  Reduce  |  to  twenty-sevenths. 

14.  Reduce  -J  to  thirds.    16.  Reduce  ^  to  fourths. 


[50.  How  reduce  a  fraction  to  any  required  denomination  ? 


REDUCTION     OF     FRACTIONS.  109 

151.  To  reduce  a  Whole  Number  to  a  Fraction  having  a 
given  Denominator, 

1.  Eeduce  7  to  fifths. 

Analysis. — ist.  Since  there  are  5  fifths  in  a  unit,  an)  number 
must  contain  5  times  as  many  Jifths  as  units ;  and  5  time&  7  are  35. 
Therefore  T—^t. 

Or,  2(1.  Since  1  =  3,7  must  equal  7  times  f, 
or  ^5^.    In  the  operation  7  is  both  multiplied  Operation. 

and  divided  by  5;  hence,  its  value  is  not     7^(7  X5)-^5=^. 
altered,     (Art.  86,  a.)     Hence,  the 

EuLE. — Multiply  the  whole  number  ly  the  given  denomi^ 
nator,  and  place  the  product  over  it. 

2.  Change  7  to  a  fraction  having  9  for  its  denominator. 

3.  Change  6:^  to  5ths.  9.  468  to  76ths. 

4.  Eeduce  79  to  7ths.  10.  500  to  87ths. 

5.  Eeduce  83  to  9ths.  11.  1560  to  hundredths. 

6.  Eeduce  105  to  i6ths.  12.  2004  to  thousandths. 

7.  Eeduce  217  to  2oths.  13.  500  to  ten-thousandths. 

8.  Eeduce  321  to  49ths.  14.  25  to  millionths. 

CASE   VI. 
152.  To  reduce  a  Complex  Fraction  to  a  Simple  one, 

I.  Eeduce  the  complex  fraction  ^  to  a  simple  one. 

Analysis. — The  denominator  of  a  fraction,  we  Operation. 

have  seen,  is  a  divisor.     Hence,  the  given  complex  •5I 

fraction  is  equivalent  to  3^-^ 5.     (Art.  142.)     Now  ~^^3"3"~^5  5 

3\=^;  therefore,  35-5=¥-5.     (Art.  148.)    But  .r^jo  . 

\^-5-  5  =z  § ;  for,  dividing  the  numerator  by  any  num-  '^  ^       ^  " 

ber,  divides  the   fraction  by  that  number.     (Art.  "T~'5— V* 

144,  Prin,  6.)     Therefore,  f  is  the  simple  fraction  A7IS.  |. 
required. 

Or,  multiplying  the  denominator  by  5,  divides  the  fraction,  and 
we  have  |f  =  |,  the  same  as  before.     (Prin.  7.) 


150.  Upon  what  principle  Is  this  rule  fonnded  ?    151.  Bow  rednce  a  whole 
number  to  a  fraction  having  a  given  denominator  ? 


Operj 

rioN. 

3 

4^ 

-i- 

^7; 

V 

i- 

-7-- 

=  2V 

A 

ns. 

A. 

UO  REDUCTIOK     OF     FRACTIONS. 

3 

2.  Reduce  the  complex  fniction  -  to  a  simple  one. 

Analysis. — Reasoning  as  before,  the  given  fraction 
is  equivalent  to  f-f-7.  But  we  cannot  divide  the 
Numerator  of  the  fraction  f  b j  7  without  a  remainder, 
we  therefore  multiply  the  denominator  by  it ;  for, 
multiplying  the  denominator,  divides  the  fraction ; 
and  4-f-7--o\-,  the  simple  fraction  required.  (Art.  144, 
Prin.  7.)     Hence,  the 

EuTiE. — I.  Reduce  the  numerator  of  tlie  complex  fraction 
to  a  simple  one. 

II.  Divide  its  numerator  hy  the  given  denominator. 
Or,  MuUijjly  its  denominator  hy  the  given  denominator. 

N-WES. — I.  When  the  numerator  can  be  divided  without  a  remain- 
der,, the  former  method  is  preferable.  When  this  cannot  be  done, 
tbe  latter  should  be  employed. 

?..  After  the  numerator  of  the  complex  fraction  is  reduced  to  a 
bnaple  one,  the  factors  common  to  it  and  the  given  denominator, 
should  be  cancelled. 

jr^"  If  this  case  is  deemed  too  difficult  for  beginners,  it  may  be 
r«waitted  till  review. 

(For  the  method  of  treating  those  expressions  which  ha^vQ  fractional 
xUnominators,  see  Division  of  Fractions,  p.  130.) 

2.  Reduce  —  to  a  simple  fraction. 

24 

Solution. — 5t=Y  ;  and  ^^--^-2^=-^^^,  or  H.  Ans. 

Rednce  the  following  complex  fractions  to  simple  ones. 

•33  108  rIO 

^*    4'  ''     4  *  '     5  * 


2f 


168  176 

8.  :^.  12.  ^^, 

■   25                        3  4 

9t                                24  59 1 

5     4                            y     7  -^     78 

24                                                 oTO  r     5  . 

6.  '^.                       10.  ^.  14.  5^^. 

2                                  2  9 

152.  How  reduce  a  complex  fraction  to  a  simple  one?    Note.  WhaX  should  be 
done  with  common  factors  ? 


EEDUCTION     OF     FEACTIONS.  Ill 


CASE     VII. 
153.  To  reduce  Fractions  to  a  Coimnon  Denominator, 

Def. — A  Comrnoii  Denominator  is  one  that 
belongs  equally  to  two  or  more  fractions  j  as,  f,  f,  -f. 

I.  Keduce  i,  f ,  J  to  a  common  denominator. 

Remake. — In  reducing  fractions  to  a  common  denominator,  it 
should  be  observed  that  the  value  of  the  fractions  is  not  to  be  altered. 
Hence,  whatev^er  change  is  made  in  any  denominator,  a  corresponding 
change  must  be  made  in  its  numerator. 

Analysis. — If  each  denominator  is  mul- 
tiplied by  all  the  other  denominators,  the 
fractions  will  have  a  common  denominator, 
and  if  each  numerator  is  multiplied  into  all 
the  denominators  except  its  own,  the  terms 
of  each  fraction  will  be  multiplied  by  the 
same  numbers ;  consequently  their  value 
will  not  be  altered.  (Art.  144.)  The  frac- 
tions thus  obtained  are  if,  ^f,  and  if. 
Hence,  the 


OPERATION. 

I 

__  ix3X4_ 

12 

2 

2x3x4 

24 

2 

2x2x4 

16 

3 

3x2x4 

24 

3 

_3X2X3  _ 

18 

4 

4x2x3 

24 

Rule. — Multipty  the  terms  of  each  fraction  Ixj  all  the 
denominators  except  its  own. 

Notes. — i.  Mixed  numbers  must  first  be  reduced  to  improper 
fracLions,  compound  and  complex  fractions  to  simple  ones. 

2.  This  rule  is  founded  upon  the  principle,  that  if  hoth  terms  of  a 
fraction  are  multiplied  by  the  same  numbers,  its  value  is  not  altered. 

3.  A  common  denominator,  it  will  be  seen,  is  the  product  of  all  the 
denominators  ;  hence  it  is  a  common  multiple  of  them.     (Art.  129.) 

Reduce  the  following  to  a  common  denominator : 


2. 

1 

andf. 

5- 

i 

i 

and^^. 

8. 

f ,  h  ih  «. 

3- 

h 

f, 

and 

f 

6. 

h 

-fj,  and  tV 

9- 

fi 

40 

T^. 

4- 

h 

i 

and 

4 

TT- 

7- 

f.^.tVandyV 

10. 

H, 

19       22 

153.  What  is  a  common  denominator?  How  reduce  fractions  to  a  common 
denominator  ?  Note.  What  is  this  mle  based  upon  ?  What  is  to  be  done  with 
mixed  numbers,  compound  and  complex  fractions  ? 


113  KEDUCTIOK     OP     FRACTIONS. 

II.  Find  a  common  denominator  of  4,  3I,  and  |  of  | 

Analysis.— 4  =  ^;  3i  =  i;  and  -^  of  f  =  J-.    Reducing  ^,  ^    i 
to  a  common  denominator,  thej  become  ^3^,  \^,  and  f . 

12.  Eeduce  3I-,  ij,  2f.  15.  Eeduce  5 J,  ^  of  8,  ^J. 

13.  Reduce  6|,  yf^,  y^^,  f.      16.  Eeduce  f  of  ^  of  9,  1 1  J,  /p 

14.  Eeduce  f  of  |,  5I,  |.       17.  Eeduce  13  J,  17,  -f  off. 

18.  Find  a  common  denominator  of  2  J,  5y3_  and  2f. 

19.  Eeduce  -^^o,  |,  and  ^J,  to  the  common  denominator 
100. 

Solution.— The  fraction -A- =fJV;  f =-Afu ;  andH=A%. 

20.  Change  J,  f,  and  f  to  72ds. 

'     21.  Change  |,  y^^,  and  j^J  to  96ths. 

22.  Eeduce  J  of  f  and  U  to  9oths. 

23.  Eeduce  |-  of  lof  and  ^^0,  to  75ths. 

24.  Eeduce  f,  -j^,  and  ^J  to  i68ths. 

25.  Eeduce  ■^,  ff,  and  y-J,  to  loooths. 


CASE    VIII. 

154.    To    Reduce    Fractions   to    the   Least    Common 
Denominator. 

Def. — A  Co7nni07i  Denoininator  is  a  common 
m?/?^ijt??e  of  all  the  denominators.     (Art.  153,  ?z.)     Hence, 

The  Least  Common  JDenominator  is  the  least 
common  multiple  of  all  the  denominators. 

I.  Eeduce  J,  y^,  and  ^  to  the  least  common  denomi- 
nator. 

Analysis. — Here   are  two  steps :    ist.  To  find  operation. 

tlie  least  common  multiple  of  the  denominators ;  $)^,  12,  15 

2d.  To  reduce  the  given  fractions  to  tliis  denomi-  "^       a      c 

nator.     The  least  common  multiple  of  the  given  •?  x  4  X  t?  —  60 
denominators  is  60.    (Art.  133.) 

154.  What  is  the  least  common  denominator?  How  reduce  fractions  to  the 
least  common  denominator  ?  Note.  What  is  to  be  done  with  mixed  numbers, 
impound  and  complex  fractions  ? 


REDUCTION     OF     FRACTIOJs^S.  113 

To  reduce  the  given  fractions  to  6oths,  we  multiply  both  terms  of 
I  by  20,  and  it  becomes  |§.  In  like  manner,  if  both  terms  of  -^^^  are 
multiplied  by  5,  it  becomes  |f  ;  and  multiplying  both  terms  of  i^  by 
4,  it  becomes  f^.  Therefore,  |,  -fj,  and  {I-  are  equal  to  |^,  |f  and 
1^.     Hence,  the 

EuLE. — Find  the  least  common  multiple  of  all  the  denom- 
inators, and  multiply  both  terms  of  each  fraction  hy  such  a 
number  as  ivill  reduce  it  to  this  denominator.    (Art.  150,  n.) 

Notes. — i.  Mixed  numbers  must  be  reduced  to  improper  frac- 
tions ;  compound  and  complex  fractions  to  simple  ones ;  and  all  frac- 
tions to  the  loicest  terms,  before  applying  the  rule.  If  not  reduced 
to  the  lowest  terms,  the  least  common  multiple  of  the  denominators 
is  liable  not  to  be  the  least  common  denominator.     (See  Ex.  2,  18.) 

2.  This  rule,  like  the  preceding,  is  based  upon  the  principle  that 
multiplying  hath  terms  of  a  fraction  by  the  same  number  does  not 
alter  its  value.     (Art.  144,  Prin.  9.) 

2.  Keduce  15,  2J,  f  of  f,  and  -^2  ^^  ^^^^  ^^ast  common 
denominator. 

I'l      T       7    2    ^S       2       ,6        I 
Analysis.— 1 1  =  ^  2^  =:  -' ,  _  of  -  r=  ~,  and  — ^  =:  -. 

^        I       '       3^      5       5  12       2 

Now    the    least     common    multiple    of    the    denominators    of 

IS    7    2    I    .  .        450    70    12    15 

^   L   -      ,  IS  30.       Ans.  -±-   -—   — y  — -i 

I  '  3    5    2  30     30    30    30 

Eeduce  the  following  to  least  common  denominator : 

3-  T?  ^'  y-  II-    -m>  TTT?  TS^  DS' 
J,      I       5        3                                     T^32I46 

4-  ^?    2(1'   T2^-  12.     y,    ^,   ^,    y,    y. 

5'hhil  13.  9h  Hi  f  of  40. 

6.  f,  f,  4i  14.  T^  of  13,  j\^,  /2V 

7.  A.  A.  f  of  i2i  15.  7|,  A  of  1 7,  il 

8.  A.  h  f  of  10.        16.  ,\%  m  uU' 

n      5      14      ^I      C3  T^       IIS      2ftQ       T447 

Tr.        9nf82      7r.f^r,  tS      15  0      2  6  5      17  2  8 

10.    Y?7  01  o?,  -g^  or  40.  15.    -yj^,  :fjjQ,  T^T5"« 


114  ADDITION    OF    FRACTIOi^S. 


ADDITION   OF   FRACTIONS. 

155.  Addition  of  Fractions  embraces  two  classes 
of  examples,  viz. :  those  which  have  a  common  denominator 
and  those  which  have  different  denominators. 

156.  When  tivo  or  more  fractions  have  a  common  de- 
nominator,  and  refer  to  the  same  kind  of  unit  or  base, 
their  numerators  are  like  parts  of  that  unit  or  base,  and 
therefore  are  like  numbers.  Hence,  they  may  be  added, 
suitracted,  and  divided  in  the  same  manner  as  whole  num- 
bers. Thus,  I  and  ^  are  12  eighths,  just  as  5  yards  and 
7  yards  are  12  yards.     (Art.  lor,  Def.  13.) 

157.  When  two  or  more  fractions  have  different  denom- 
inators, their  numerators  are  unlike  parts,  and  therefore 
cannot  be  added  to  or  subtracted  from  each  other  directly, 
any  more  than  yards  and  dollars.     (Art.  10 1,  Def.  14.) 

158.   To  add   Fractions  which   have  a   Common  Denom.- 
inator. 

1.  What  is  the  sum  of  f  yard,  |  yard,  and  |  yard  ? 

Analysis.— 3  eigliths  and  5  eighths  operation. 

yard  are  8  eighths,  and  7  are  15  eighths,     f  + 1  +  -g-—  ¥>  ^r  i|  y. 
equal  to  i^  yard.     (Art.  147.)    For,  since 

the  given  fractions  have  a  common  denominator,  their  numerators 
are  like  numbers.     Hence,  the 

EuLE. — Add  the  numerators,  and  jflace  the  sum  over  the 
common  denorninator. 

Note. — The  answers  should  be  reduced  to  the  lowest  terms;  and  If 
improper  fractions,  to  whole  or  mixed  numbers. 

Add  the  following  fractions . 

2.  -f-,  f,  and  f  5.  ^^,  ^,  T^,  and  «*. 
3-  A,  A,  A,  and  «.  6.  ^^i^,  ^6^,  «$,  and.||f. 
4.  ^,  A,  ^,  and  H-  7.  Ml.  Hh  lif.  and  i|l 

158.  How  add  fractioUB  that  have  a  com^ion  denominator? 


ADDITIOi^     OF     FU  ACTIONS.  115 

159.  To  add  Fractions  which  have  Different  Denomina- 

tors, 

8.  Find  the  sum  of  J  of  a  pound,  f  of  a  pound,  and  |  of 
a  pound. 

Analysis.— As  these  fractions  operation. 

have      different     denominators,  4  x  5  X  8  =  160,  com.  d, 

their  numerators  cannot  be  added  1x5x8=    40,  ist  nu. 

in  their  present  form,    (Art.  157.)  2x4x8=    64,  2d     " 

We  therefore  reduce  them  to  a  5x4x5  =  100,  3d     " 
common  denominator,  which  is          g^^ni  of  nu.,  204.     Hence, 

160.  Then  \=-,^^,  |=-,Vo,  and     ^q        5^       ioo_204 

§ = -\M.    Adding  the  numerators,     — 7"  +  "7~  +  "~7~  —  "^7 ' ,  or  I  ^^ 
"     ''^'      .        ^,^  ^,        160      160      160      160 

and  placing  the    sum  over   the 

common  denominator,  we  have  }^^,  or  i^^,  the  answer  required. 

For,  reducing  fractions  to  a  common  denominator  does  not  alter 

their  value ;  and  when  reduced  to  a  common  denominator,  the  nu 

merators  are  like  numbers.     (Arts.  153,  156.) 

Or,  we  may  reduce  them  to  the  least  common  denominator,  which 

is  40,  and  then  add  the  numerators.     (Art.  154.)    Hence,  the 

KuLE. — Reduce  the  fractions  to  a  common  denominator, 
and  place  the  sum  of  the  numerators  over  it. 

Or,  reduce  the  fractions  to  the  least  common  denomina- 
tor, and  over  this  place  the  sum  of  the  numerators. 

Remarks. — i.  The  fractional  and  integral  parts  of  mixed  num- 
bers should  be  added  separately,  and  the  results  be  united. 

Or,  wJiole  and  mixed  numbers  may  be  reduced  to  improper  frac- 
tions, then  be  added  by  the  rule.    (Art.  148.) 

2.  Compound  and  complex  fractions  must  be  reduced  to  simple 
ones ;  then  proceed  according  to  the  rule.     (Ex.  20,  28.) 

(For  Addition  of  Denominate  Fractions,  see  Art.  315.) 

9.  What  is  the  sum  of  5,  2  J,  and  io|  ? 

Analysis, — Reducing  these  fractions  to  a  common       5    =5 
denominator  12,  we  have,  5  =  5  ;  2i=::2-,4,;  and  io|=        24  =  2-1^7 
iOi^2,      Adding  the  numerators,  4.  twelfths    and  9 
twelfths  are  if— i-i\.    i  and  10  are  11  and  2  are  13  and 
5  are  18.     Ans.  iSff.    Or,  5=f  or  ^;  2^=:M;  and 
io|= J-i^a     Now  ft  +  f  I  +  W= W,  or  i8f,-  Ans. 

159,  How  when  they  have  not  ?  Note.  How  add  whole  and  mixed  numbers  ? 
Compound  and  complex  fractions  ? 


T2 


116 


.6 

ADDITION 

OF   feactio:ms 

• 

Add  the  following : 

(10.) 

(11.) 

(12.)           (I3-) 

(14.? 

2l 

3A 

4l 

Si 

"1 

27A           H 

8A           U 

461              li 

(IS-) 
I9i 
47i 
68f 

(i6.) 

68A 

(17.)           (18.) 

2o7i         i7Sl 
62|-          207 

49A    368J 

('9-) 

45° 
67^ 
3715 

20.  What  is  the  sum  of  J  of  J,  f  of  7,  and  ^  of  9I  ? 

21.  What  is  the  sum  of  f  of  f,  f  of  |,  and  f  of  7  ? 

22.  What  is  the  sum  of  f  of  4 J,  f  of  3,  and  f  of  10  J  ? 

23.  A  beggar  received  $if  from  one  person,  $2^  from 
another,  and  $3 J  from  another :  how  much  did  he  receive 
from  all  ? 

24.  If  a  man  lays  up  $43!  a  month,  and  his  son  I27I, 
how  much  will  both  save  ? 

25.  What  is  the  sum  of  f ,  |,  -J J,  and  f  pound  ? 

26.  A  shopkeeper  sold  i  j^  yards  of  muslin  to  one  cus- 
tomer, 8 J  yards  to  another,  2 5 -J  to  another:  how  many 

\  yards  did  he  sell  to  all  ? 

\  „  27.  A  farmer  paid  |i8J  for  hay,  $45t\  for  a  cow,  $150! 

for  a  horse,  and  $275  for  a  buggy:  how  much  did  he  give 

for  all? 

28.  What  is  the  sum  of  — ^,  — ,  and  t 

5      3  4 

Analysis. — Rt  lucing  the  complex  fractions  to  simple  ones,  we 

have,  -4i=|-5=:,%  ;  ^i  =  \^^3=i  ;  and  ^=l-^4=-h,  or  i     Now, 
53  4 

fp  l-l  +  i-t^  +  \¥  +  ^^  =  2a^l^«- 

48A6  »72I  «2      140  A3 

29.  Add  '^,  ^,  and  ^-^A    30.  Add  ^,  =f^,  and  ^. 
^23  7  573 


SUBTRACTION     OF     FRACTIONS.  117 


SUBTRACTION    OF    FRACTIONS. 

160.  Subtraction  of  Fractions  embraces  tut 
classes  of  examples,  viz.:  those  which  have  a  common 
denominatory  and  those  which  have  different  denominators. 

161.  To    subtract    Fractions   which    have    a    Common 

Denominator, 

I.  What  is  the  difference  between  |f  and  \^  ? 
Analysis. — 13  sixteenths  from  15  sixteentlis 

-  .  'Ti  .1  •       J        T-.  OPERATION. 

leave  2  sixteenths,  the   answer  required.     For, 

T  C  T  •J  2 

since  the  given  fractions  have  a  common  denomi-     _:?. •£  ^^ .^ 

nator,  their  numerators,  we  have  seen,  are  like     16        16        16 
numbers.    (Art.  156.)    Hence,  the 

EuLE. — Take  the  less  numerator  from  the  greater',  a7id 
place  the  difference  over  the  common  denominator. 


2 


From  if  take  \%.  5.  From  J-Ji  take  -Jf  J 

3.  From  fl  take  f f.  6.  From  |4|  take  %%%. 

4.  From  \l\  take  ^V  7-  From  ^^/o  take  ^«oV 

162.    To  subtract   Fractions  which   have  Different 
Denominators, 

8.  From  f  of  a  pound,  subtract  j  of  a  pound.        6  _  24 

,    Analysis. — Since    these    fractions    have    different  7        28 

denominators,  their  numerators  cannot  be  subtracted  3        2 1 

in  their  present  form.     We  therefore  reduce  them  to  a  a  ~  28 

common  denominator,  which  is  28  ;  then  subtracting  as  

above,   have    2^8.  the    answer    required.     (Art.    153.)  Ans.    — 

Hence,  the  28 

Rule. — Reduce  the  fractions  to  a  common  denominator, 
and  over  it  place  the  difference  of  the  numerators. 

161.  How  subtract  fractions  that  have  a  common  denominator?  162.  How 
when  they  have  different  denominators  ?  Bern.  How  subtract  mixed  numbers  \ 
Compound  and  complex  fractions  ?    A  proper  fraction  from  a  whole  number? 


118  SUBTRACTIOi^    OF     FRACTIOIn^S. 

Remarks. — i.  HhQ  fractional  and  integral  parts  of  mixed  numbers 
should  be  subtracted  separatelj,  and  the  results  be  united. 

Or,  they  may  be  reduced  to  improper  fractions,  then  apply  the  rule. 

Compound  and  complex  fractions  should  be  reduced  to  simple  ones, 
and  all  fractions  to  their  lowest  terms. 

2.  A.  proper  fraction  maybe  subtracted  from  a  wliole  number  by 
taking  it  from  a  unit;  then  annex  the  remainder  to  the  whole 
number  minus  1. 

Or,  the  whole  numher  may  be  reduced  to  a  fraction  of  the  same 
denominator  as  that  of  the  given  fraction  ;  then  subtract  according 
to  the  rule.     (Art.  151.) 

3.  The  operation  may  often  be  shortened  by  finding  the  least  com 
mon  denominator  of  the  given  fractions.     (Art.  154.) 

(For  subtraction  of  Denominate  Fractions,  see  Art.  317.) 

9.  What  is  the  difference  between  1 2}  pounds  and  5^ 
pounds  ? 

AiTALYSTS. — The  minuend  I2^=i2f.  Now  i2|— 5|=6|,  Ans. 
Or,  12^-^5;^  or^ii;  and  5!  =  ^^.  Now^4^— \^=^/,  which  reduced 
\o  a  mixed  number,  equals  6|,  the  same  as  before. 

(10.)         (II.)  (12.)  (13.)  (14.) 

^  .        From     I        .    5 1  7i-  23I   .         ^ 

U-^      Take      t      "^     z\  5 1      ^      ^Sf  li 

(15.)         (16.)        (17.)         (18.)         (19.) 
From     5*1  7ii  ^\%  8^  9t| 

Take     3if  4H  3lf  SM  7t'A 

« 

20.  From  a  box  containing  56^  pounds  of  sugar,  a  grocer 
took  out  23  J  pounds :  how  many  pounds  were  left  in  the  box? 
'"  ,  21.  From  a  farm  containing  165^^^  acres,  the  owner  sold 
*    78 J  acres:  how  much  land  had  he  left? 

22.  From  13  subtract  f. 

Analysis. — ist.  Reducing  13  to  sevenths,  we  have  i3=-f,  and 
3,^—5  =  87^,  or  I2f  Ans. 

Or,  2d.  Borrowing  i  from  13,  and  reducing  it  to  7th s,  we  have 
13  =  12?^,  and  12^— ^=i2f,  An». 


SUBTRACTION     OF     FRACTIONS.  119 

23.  From  46  take  7  J.  24.  From  58  take  2o|. 

25.  From  84f  take  41.  "/  2S.  From  150I  take  8^. 

l^  27.  From  no  take  7-AV  28.  From  1000  take  999f  J 

29.  Subtract  J  of  J  of  3  from  ^  of  f  of  i^. 
Analysis.— Reducing     i        4  j_i       4    i.3_3_£^ 

the  compound  fractions      ^  ^    t  ^    ^^~2.^    K        2.~  K  ~20 
to  simple  ones,  then  i^Ojj  IlSi"? 

a  common  denominator,     -  of  -  of  3    =  -  of  -  of  -  =  -  ==  ^ — - 
the  subtrahend  becomes     3       4  ^4^4        ^^ 

is,    the    minuend    \l.  .  7^ 

Nowi§-A  =  /o.    Ans.  ^-^^*-  20* 

(30.)  (31')  (32')  (33*) 

From    foff  |ofii  Aof4i  ift  of  28 

Take     iof|  f  of  A  f  of  3  |  of  4! 

34.  From  ^  of  62^  subtract  ^  of  16  J. 

35.  A's  farm  contains  256!  acres,  B's  43ixV  acres:  what 
is  the  difference  in  the  size  of  their  farms  ? 

36.  If  from  a  yessel  containing  230^^  tons  of  coal,  119^ 
tons  are  taken,  how  much  will  there  be  left  ? 

37.  A  man  bought  a  cask  of  syrup  containing  58^ 
gallons;  on  reaching  home  he  found  it  contained  only  17I 
gallons :  how  many  gallons  had  leaked  out  ? 

38.  A  lady  having  $100,  paid  $8 J  for  a  pocket  hand- 
kerchief, $15^  for  a  dress  hat,  $46!  for  a  cloak:  how 
much  had  she  left  ? 

39.  From  ^^  subtract  — . 
'^^  6  5 

Analysis.— Reducing  the  complex  5I.     j,.^      jj      5- 

fractions  to  simple  ones,  then  to  a  ~^^^^"~^^'~T2—'5% 

common  denominator,  the  minuend  j 

becomes  fi  the  subtrahend  VS.  (Arts.  -^=^^^=^-^=z    I-6 

152,    153.)      Now,   f^-U  =  H,  or  5 


\l,Ans.  Ans.^—2^ 

40.  From  --^  take  t-  4i.  From  ^  take  ^i 

^  t:  8  ^22 


120        MULTIPLICATION     OF     FRACTIOXS. 

MULTIPLICATION    OF    FRACTIONS. 
CASE    I. 

163.  To  multiply  a  Fraction  by  a  Whole  Kuniher, 

Ex,  I.  What  will  4  pounds  of  tea  cost,  at  |  of  a  dollar  8 
pound  ? 

1st  Method. — Since  i  pound  costs  $|,  4  pounds        ist  operation. 
wiU  cost  4  times  as  much,  and  4  times  $|  are  $|-X4=$-^ 

\tt— I2L,  whicli  is  the  answer    required.      For,     and  $^^=$2^ 
multiplying  the  numerator,  multiplies  the  frac- 
tion.   (Art.  144,  Prin.  5.) 

2d  Method. — If  we  divide  the  given  de-  2d  operation. 

nominator  by  the  given  number  of  x>ounds,     $|-r-4  =  $|,  or  $2^ 
we  have  1-^-4= $f,  or  $22,  which  is  the  true 

answer.      For,   dividing  the  denominator,   multiplies  the  fraction. 
(Art.  144,  Prin.  8.)    Hence,  the 

EuLE. — Multiply  the  numerator  hy  the  whole  nurriber. 
Or,  divide  the  denominator  hy  it. 

Remabks. — I.  When  the  multiplicand  is  a  mixed  number,  the 
fractional  and  integral  parts  should  be  multiplied  separately,  and 
the  results  be  united. 

Or,  the  mixed  number  may  be  reduced  to  an  improper  fraction, 
and  then  be  multiplied  as  above. 

2.  K  a  fraction  is  multiplied  by  its  denominator,  the  product  will 
be  equal  to  its  numerator.  For,  the  numerator  is  both  multiplied 
and  divided  by  the  same  number.    (Art.  86,  a.)    Hence, 

3.  A  fraction  is  multiplied  by  a  number  equal  to  its  denominator 
by  cancelling  the  denominator.     Thus,  f  x  8  =  5.     (Art.  121.) 

4.  In  like  manner,  a  fraction  is  multiplied  by  any  factor  of  its  de- 
nominator by  cancelling  that  factor. 

2.  Multiply  2  7f  by  6. 

Analysis. — Multiplying  the  fraction  and  integer  operation. 
separately,  we  have  6  times  f=-S^,  or  3I ;   and  6  27^ 

times  27=162.     Now  162  +  31=1651,  the  answer.  ^ 

Or,thus:27|=H^;andH^x6=«F^ori65l,  ^ns.  Ans.   165I 

163,  How  multiply  a  fraction  by  a  whole  number  ?  Upon  what  does  the  first 
method  depend?  The  second?  Rem.  How  multiply  a  mixed  number  by  a 
whole  one  ?  How  multiply  a  fraction  by  a  number  equal  to  its  denominator  f 
How  by  any  factor  in  its  denominator  ? 


Ml 

[JLTIP 

LICATIi 

ON     OF 

ERACTIONB. 

(3-) 

(4.) 

(5.) 

(6.) 

(7.) 

Mult. 

If 

a 

23J 

35i 

48i 

By 

_7 

_9 

8 

10 

12 

(8.) 

(9.) 

(10.) 

(II.) 

(12.) 

Mult. 

n 

U 

Ui 

98A 

ssifV 

By 

48 

100 

78 

26 

48 

12i 


13.  Multiply  lif  by  239. 

14.  What  cost  8  barrels  of  cider,  at  1;^  a  barrel? 

15.  At  $25!:  each,  what  will  9  chests  of  tea  cost? 

16.  What  will  25  cows  come  to,  at  I48J  apiece? 

17.  What  cost  27  tons  of  hay,  at  $29^  a  ton? 

18.  What  cost  35  acres  of  land,  at  $45  J  per  acre  ? 

19.  At  $34 J  apiece,  what  cost  50  hogsheads  of  sugar. 

20.  At  $45f  apiece,  what  will  100  coats  come  to  ? 

21.  What  cost  6  dozen  muffs,  at  $7 5  J  apiece  ? 

CASE  II. 

165.  To  Multiply  a   JVhole  Kimiher  by  a  Fraction. 

Def.  Multiplying  by  a  Fraction  is  taking  a 
sertain  part  of  the  multiplicand  as  many  times  as  there 
are  UTce  parts  of  a  unit  in  the  multiplier. 

But  to  find  a  given  part  of  a  number,  we  divide  it  into 
as  many  equal  parts  as  there  are  units  in  the  denominator, 
and  then  take  as  many  of  these  parts  as  are  indicated  by 
the  numerator.     (Art.  143.)     That  is, 

To  multiply  a  number  by  ^,  divide  it  by  2. 

To  multiply  a  number  by  ^,  divide  it  by  3. 

To  multiply  a  number  by  J,  divide  it  by  4  for  J,  and 
multiply  this  quotient  by  3  for  f,  etc.  Hence, 

Remarks. — i.  Multiplying  a  whole  number  by  a  fraction  ig 
the  same  as  finding  a  fractional  part  of  a  number,  which  the  pupil 
should  here  review  with  care.    (Art.  143.) 

I)3f.  What  is  it  to  multiply  by  a  fraction  ?   How  multiply  by  i  ?   By  i  ?  By  1 1 


122  MULTIPLICATION?^    OF    FEACTIONS. 

2.  When  the  multiplier  is  i,  the  product  is  equal  to  the  multipli- 
cand. 

When  the  multiplier  is  greater  than  i,  the  product  ia  greater  than 
tlie  multiplicand. 

When  the  multiplier  is  less  than  i,  the  product  is  less  than  the 
multiplicand. 

1.  What  will  J  of  a  ton  of  iron  cost,  at  $55  a  ton  ? 

ist  Method. — Since  i  ton  costs  $55,  |  of  a  ton  opekatiok. 

will  cost  f  times  $55,  or  f  of  $55.     But  f  of  $55  H)S5 

=  3  times  i  of  $55.     Now  \  of  $55==$55-^4,  or  $133 

$13!  ;  and  3  times  $i3l=$4i4,  the  answer  required.  3 

For,  by  definition,  multiplying  by  a  fraction  is  a  g  ^71^ 
taking  a  part  of  the  multiplicand  as  many  times  as 
there  are  like  parts  of  a  unit  in  the  multiplier.  Now  dividing  the 
whole  number  by  4,  takes  i  fourth  of  the  multiplicand  once  ;  and 
multiplying  this  result  by  3,  repeats  this  part  3  times,  as  indicated  by 
the  multiplier.     (Art.  143.) 

2d  Method.— I  of  $55=^  of  3  times  I55.    But  $55 

$55  X  3  =  $165,   and   §i65-T-4— $414,  the   same  as  2 

before.    For,  i  of  3  times  a  number  is  the  same  as  /i')76c^ 

3  times  4-  of  it.    Hence,  the  .      -.      .,- 

^          *  Ans.  $4ii 

Rule. — Divide  the  whole  numher  hy  the  denominator  of 
the  fraction,  and  multiply  hy  the  numerator. 

Or,  Multiply  the  whole  numher  hy  the  numerator  of  the 
fraction,  and  divide  hy  the  denomi?iator. 

Remarks. — i.  The  fraction  may  be  taken  for  the  multiplicand, 
and  the  whole  number  for  the  multiplier,  at  pleasure,  without  affect- 
ing the  result.     (Art.  45.) 

2.  When  the  multiplier  is  a  mixed  number,  multiply  by  the  frac- 
tional and  integral  parts  separately,  and  unite  the  results. 

Or,  reduce  it  to  an  improper  frax^tion ;  then  proceed  according  to 
the  rule.     (Ex.  2.) 

2.  What  will  8f  tons  of  copper  ore  cost,  at  $365  a  ton  ? 

Analystb.— 8|  =^^  ;  now  ^/  tons  will  come  to  ^i-  times  $365  ;  and 
$365  X  Y=$3i39-  Ans. 

165.  How  is  a  whole  number  multiplied  by  a  fraction?  Eem.  WbaX,  is  the 
operation  lilce ?  When  the  multiplier  is  i,  what  ia  the  product  ?  When  the  mul- 
tiplier is  greater  than  i,  what  ?  When  lesa,  what?  Re.n  IIow  multiply  a  whole 
by  a  mixed  number? 


MULTIPLICATIOIf     OF     FRACTIONS.         123 


(3-) 

(4.) 

(5-) 

{6., 

(7.) 

Mnlt. 

65 

89 

96 

87 

100 

By 

i 

^ 

3i: 

4j 

5f 

8.  What  cost  ^  of  an  acre  of  land,  at  $38  an  acre  ? 

9.  At  $29  a  ton,  what  cost  8|  tons  of  hay  ? 

10.  "What  cost  2  8f  tons  of  lead,  at  $223  per  ton  ? 

11.  What  is   fjof    845?  15.  Multiply   79  by  yf. 

12.  W^hat  isy'^  of  1876  ?  16.  Multiply  103  by  gf. 

13.  What  is   If  of  1000?  17.  Multiply  1 00 1  by2iyV 

14.  What  is  J-J§  of  2010  ?  18.  Multiply  1864  by  37-^. 

CASE   III. 
166.  To  multiply  a  Fraction  by  a  Fraction. 

I.  What  will  f  of  a  pound  of  tea  cost,  at  $^  a  pound  ? 

Analysis. — \  of  a  pound  will  cost  i  sixth  operation. 

as  much  as  i  pound;  and  ^  of  $,'0-  is  $o%-  "HT  ><  |— ts"?  ^^^  %i 
Again,  ^  of  a  pound  will  cost  5   times   as  by  cancellation, 

much  as  ^;  and  5  times  %ia  are  $6B=$4,  3>  ^  ^    -'  "^^^  ^3 

the  answer  required.  2,  IBt        5,  2 

Or,  having  indicated  the  multiplication,  canrd  the  common  fac- 
tors 3  and  5,  and  then  multiply  the  numerators  together  and  the  de- 
nominators ;  the  result  is  $i^,  the  same  as  before 

The  reason  is  this:  Multiplying  the  denominator  of  the  multipli- 
cand 1^0  by  6  the  denominator  of  the  multiplier,  \S\q  result,  $,^„,  is 
5  times  too  small ;  for,  we  have  multiplied  by  ^  instead  of  ^  of  a 
unit,  the  true  multiplier.  To  correct  this,  we  must  multiply  this  re- 
sult by  5,  which  is  done  by  multiplying  its  numerator  by  5.   (Art.  144.) 

The  object  of  cancelling  common  factors  is  twofold:  it  shortens  the 
operation,  and  gives  the  answer  in  the  lowest  terms.    Hence,  the 

KuLE. — Cancel  the  common  factors  ;  then  multiply  the 
numerators  together  for  the  neio  numerator,  and  the  dejiom- 
inators  for  the  new  denominator.     (Art.  144,  Prin-  9.) 


166.  How  multiply  a  fraction  hy  a  fraction  ?   Object  of  cancelling  ?    E^m.  ITow 
multiply  compound  fractloau  ?    Mixed  numljers  !    Coinplex  fractions  ? 


124       MULTIPLICATIOK     OF     FBaCTIOI^^S. 

Remarks. — i.  Mixed  numbers  should  be  reduced  to  improper 
fractions,  and  complex  fractions  to  simple  ones ;  then  apply  the  rule. 

2.  Multiplying  by  a  fraction,  reducing  a  compound  fraction  to  a 
simple  one,  and  finding  a  fractional  part  of  a  number,  are  identical 
operations.     (Arts.  143,  149,  166.) 

2.  Multiply  f  of  J  of  f  by  i  of  4. 

SoLtmON. — It  is  immaterial  as  to  the  result,  whether  3i 

the  fractions  are  arranged  in  a  horizontal  or  in  a  per-  i^ 

pendicular  line,  with  the  numerators  on  the  right  and  6 

the  denominators  on  the  left.    (Art.  122,  Mem.)  2. 


5=^ 


Perform  the  following  multiplications : 


3- 

7- 

ifx 

4. 

Ax  A 

8. 

iix 

5- 

nx^ 

9- 

^x 

6. 

«x    f 

10. 

fix 

TT  II-  tX     ^X     y 

fy  12.  |Xt2_x     f 

TT  I3*  TXT  X -g^XT  X -4:^ 

TT  14-  T^XyyX^-j 

15.  Multiply  f  of  f  of  25  by  I  of  -§^  of  j. 

16.  Multiply  I  of  tV  of  33i  by  |  of  if  of  i8f. 

17.  If  I  quart  of  cherries  costs  J  of  J  of  45  cents,  what 
will  f  of  T^  of  a  quart  cost  ? 

1 8.  What  will  5^  barrels  of  chestnuts  cost,  at  6  J  dollars 
ft  barrel?  Ans.  $$6. 

(19.)  (20.)  (21.)  (22.) 

Mult.  24I  37-1  62J  165-J 

By  _8|-  _^  jH  _92| 

23.  Wbat  is  product  of  ~  multiplied  by  — . 

5^     21      .     21         2}    8  8 

Analysis. — -^=  — =-6  =  — ,  and  -^  —  — ~3  =  ~. 

64  24  3      3  9 

7,21     8.  7        7 

Cancelling  common  factors,  we  have     i^r;  X  -      =: =^-,Ans, 

3.^^     ^.3     3x3     9 

24.  Multiply  ^  by  ^t  25.  Multiply  ^5  by  ^. 


MULTIPLICATIOiq^     OF     FRACTIONS.        125 

167.  The  preceding  principles  may  be  reduced  to  the 


following 


GENERAL    RULE. 


I.  Reduce  tvhole  and  mixed  numlers  to  improper  frac- 
tions, and  coynplex  fractions  to  simple  07ies. 

II.  Cancel  the  common  factors,  and  place  the  product  of 
the  numerators  over  the  product  of  the  denominators. 

EXAMPLES. 

1.  What  cost  I:  of  a  yard  of  calico,  at  $J  a  yard  ? 

2.  What  cost  f  of  a  pound  of  nutmegs,  at  $f  a  pound  ? 

3.  At  $f|  a  gallon,  what  will  J  of  a  gallon  of  mokssca 
cost? 

4.  Multiply  23  J  by  6}.  7.  Multiply  pi-f  by  43^. 

5.  Multiply  42-^  by  8f .     '  8.  Multiply  i64f  by  75xV 

6.  Multiply  65I  by  <)\.  9.  Multiply  2oof  by  86^^ 

10.  What  is  the  product  of  f  of  |  of  f  of  f  into  f  of  ^^ 

ofA? 

11.  What  is  the  product  of  f  of  3I  into  \  of  19J  ? 

12.  When  freight  is  $i|  per  hundred,  what  will  it  cog* 
to  transport  27  hundredweight  of  goods  from  New  Yorl' 
to  Chicago  ? 

313.  What  will  16 J  quarts  of  strawberries  come  to,  at 
26J  cents  per  quart  ? 

14.  At  $3f  a  barrel,  what  will  be  the  cost  of  47!  barrels 
of  cider  ? 

15.  Bells  are  composed  of  J  tin  and  f  copper:  how 
many  pounds  0.  each  metal  docs  a  church-bell  contain 
which  weighs  3750  pounds  ? 

jL^  16.  What  will  45  men  earn  in   15}  days,  if  each  earns 
''$2j  per  day? 

17.  How  many  feet  of  boards  will  a  fence  603^  rods 
long  require,  allowing  74 J  feet  of  boards  to  a  rod  ? 

167.  What  is  the  general  rule  for  multlpljdng  fractions  ? 


12G  DIVISION     or     FRACTIOKS. 

DIVISION    OF    FRACTIONS. 
CASE    I. 
168.  To  divide  a  Fraction  by  a  Wliole  Number, 

1.  If  4  yards  of  flannel  cost  If,  what  will  i  yard  cost? 
ist  Method. — i  yard  will  cost  i  fourth  as 

much  as  4  yards  ;  and  dividing  the  numerator  ^®*  method. 

by  4,  we  have  $|h-4=^$|,  the  answer  required.  ^~A-  __  ^2  ^^5; 

For,  dividing  the  numerator  by  any  number,  9 
divides  the  traction  by  that  number. 

2d  Method. — Multiplying  the  denominator  2d  method. 

by  the   divisor  4  (the  number  of  yards),  the  8      8 

result   is    .^^  =  |,   the   same  as   before.     For,  9x4       36 

multiplying  the  denominator  by  any  number,  g 

divides  the  fraction  by  that  number.    (Art.  144,  Ans.  —  =  $^ 

Prin.  7.)    Hence,  the  ^ 

EuLE. — Divide  the  numerator  hy  the  whole  number. 
Or,  multiply  the  denominator  hy  it. 

Remaek. — When  the  dividend  is  a  mixed  number,  it  should  be 
reduced  to  an  improper  fraction  ;  then  apply  the  rule. 

2.  Divide  %\  by  9.     Ans.  $f -^9=Vt- 
Perform  the  following  divisions : 

/,      1 6    •    *  *T      I50_i_.or'  TT      4060    ♦    fsr\ 


4.  -u 

C      49 

5-  1^ 


r,  5      268    •    A>j  x'7      52326_j!_Tr» 

«T  ^      350_:_^p,  T-?      60045    •    t  C 


6.  |f--33-     .        10.  flf-8i.  14.  fH-m-91. 

15.  A  man  having  J  of  a  barrel  of  flour,  divided  it 
equally  among  5  persons :  what  part  of  a  barrel  did  each 
receive  ? 

16.  If  12  oranges  cost  IxVtt'  what  will  i  orange  cost? 

17.  If  7  writing-books  cost  $f,  what  will  that  be  apiece  ? 

18.  If  6  barrels  of  flour  cost  I45I,  what  will  i  barrel  cost? 


168.  How  divide  a  fraction  by  a  whole  number?  Upon  what  does  the  flrpt 
method  depend?  The  second?  Which  is  preferable?  Eem.  How  is  a  mised 
number  divided  by  a  whole  one  ? 


DIVISIO:?^     OF     FRACTIONS.                    127 

Perform  the  following  divisions : 

19.   8ox^3:~l2. 

23.    865^-82. 

27.  1000^^^50. 

20.  762V^i4. 

24.    490/0-^40. 

28.  46841^68. 

21.    28f-^20. 

25.  758if-48. 

29.  78961-^-25. 

22.    511-^27. 

26.  97511-^63. 

30.  9684^^84. 

31.  If  5  yards  of  cloth  cost  $42|,  what  will  i  yard  cost. 

32.  If  6  horses  cost  $7561,  what  will  i  horse  cost? 
^^.  If  45  lbs.  of  wool  cost  $5 4 J,  what  will  i  lb.  cost  ? 

CASE    II. 
168.  «•    To  divide  a  Whole  Kinnher  by  a  Fraction. 

I.  At  S|  a  yard,  how  many  yards  of  cashmere  can  you 
buy  for  820  ? 

Analysis.— At  %\  &  yard,  $20  will  operatiok. 

buy   as    many   yards    as    there    are  20x4=180  y.  at  %\. 

fourths  in  $20,  and  4  times  20  are  80.  8o^3=:26f  y.  at  %%. 

But  the  price  $J,  is  3  times  as  much  Or,  20  X  |=26|-  y.  at  $J. 
as  $^;  therefore,  at  $f,  you  can  buy 

only  ^  as  many  yards  as  at  %\ ;  and  ^  of  80  yards  is  26|  yards, 
the  answer  required.  For,  multiplying  the  whole  number  by 
the  denominator  4,  reduces  it  to  fourths,  which  is  the  same  denomi- 
nator as  the  given  divisor.  But  when  fractions  have  a  common 
denominator,  their  numerators  are  like  niimhers ;  therefore,  ona 
may  be  divided  hy  the  other,  as  whole  numbers.  (Art.  156.)  But 
multiplying  the  whole  number  20,  by  the  denominator  4,  and  dividing 
the  product  by  the  numerator  3,  is  the  same  as  inverting  the 
fractional  divisor,  and  then  multiplying  the  dividend  by  it.   Hence,  the 

Rule. — Multiply  the  luliole  numher  hy  the  fraction  in- 
verted.    (Art.  165.) 

Remarks. — i.  When  the  divisor  is  a  mixed  number  it  should  be 
reduced  to  an  improper  fraction  ;  then  divide  by  the  rule.    (Ex.  16.) 

2  A  fraction  is  inverted,  when  its  terms  are  made  to  exchange 
places.     Thus,  f  inverted  becomes  f . 

3  After  the  denominator  is  inverted,  the  common  factors  shouIJ 
be  cancelled.    (Art.  149,  n.) 

168,  a.  How  divide  a  whole  number  by  a  fraction  ?    How  does  it  appear  that 
t'cvis  process  will  give  the  true  answer  ? 


128  DIYISIOJf    OF    fractio:n"S. 

Perform  the  following  divisions : 

2.  95-1-  5-  i75^A-  8.  576^^. 

3.  i68-^tV  6.  26I-^^7.  9.  1236-^^. 

4-  245 -^iiF-  7-  34S-^f  10.  624o-^AV 

II.  How  many  yards  of  muslin,  at  $|  a  yard,  can  be 
bought  for  $19  ? 

Analysis. — At  $^  a  yard,  $19  will  buy  as  many  yards  as  there 
Bxe  thirds  in  19,  and  19  x  ^=57.    Therefore  $19  will  buy  57  yards. 

12.  27=how  many  times  ^  ?     14.  38=rhow  many  times  -f? 

13.  5 3= how  many  times  ^  ?     15.  67  =  how  many  times  ^  ? 

16.  How  many  cloaks  will  45  yards  of  cloth  make,  each 
containing  4^  yards  ? 

AiSTALYSis. — Keducing  the  divisor  to  an  improper  fraction,  we  have 
41^=^,  and  45  yds.  x  $=^9^,  or  10  cloaks.,  Ans. 

17.  Divide  88  by  lof.  19.  Divide  785  by  62}. 

18.  Divide  100  by  12 J.  20.  Divide  1000  by  Sj-}. 

CASE   III. 

169.  To  divide  a  Fraction  by  a  Fraction,  when  they  have  a 
Common  I>enorninator. 

Eemark. — This  case  embraces  two  classes  of  examples: 
First,  those  in  which  the  fractions  have  a  common  denominator. 
Second,  those  in  which  they  have  different  denominators. 

21.  At$f  apiece,  how  many  melons  can  be  bought  for  $J? 

Analysis. — Since  $|  will  buy  i  melon,  %l  will  buy    operation. 
fts  many  as  $1   are  contained  times  in   $| ;    and   3     l~^f=^^i 
eighths  are  in  7  eighths,  2  times  and  i  over,  or  2^  times,  Ans,  2J  m. 
(Art.  156.)     Hence,  the 

EuLE. — Divide  the  numerator  of  the  dividend  })y  that 
of  the  divisor. 

Note. — When  two  fractions  have  a  common  denominator,  their 
numerators  are  like  numbers,  and  the  quotient  is  the  same  as  if  they 
were  whole  numbers.    (Arts,  64,  156.) 

22.  Divide  f|  by  ^.  24.  Divide  f|  by  J  J. 

23.  Divide  fl^yji' 25.  Divide  j|  by  jf. 

169.  IIow  divide  a  fraction  by  a  fraction  when  they  have  a  common  denominator? 


DIVISION    OF    FEACTIOJSTS.  12S 

170.  To  divide  a  Fraction  by  a  Fraction,  when  they  have 
Different  Denominators, 

1.  At  8J  a  pound,  how  much  tea  can  be  bought  for  $f  ? 
1ST  Analysis. — $f  will  buy  as  many  pounds  -|=;|-=^9^ 

as  $1  are  contained  times  in  $i     $t  =  $t.     Re-  f  =  A 

ducing  f  and  |  to  a  common  denominator,  they  9  _^  g  -—  q  .^l.  g 

become  -i%-  and  1%,  and  their  numerators  like  mim-  "^  '  J^    ^    ' 

bers.    Hence,  A-r-,%=9-r- 8, or  li  ^m.  li pound.  9-^^— ^^^"^ 

Remarks. — i.  In  reducing  two  fractions  to  a  common  denominO' 
tor,  we  multiply  the  numerator  of  each  into  the  denominator  of  the. 
other,  and  the  two  denominators  together.  But  in  dividing,  no  use 
is  made  of  the  common  denominator ;  hence,  in  practice,  multiply- 
ing the  denominators  together  may  be  omitted.     (Art.  153.) 

2D  Analysis.— $f=$!l.  At  $|  a  pound,  $1  will  buy  as  many 
pounds  as  $|  are  contained  times  in  $1.  Now  i-i-^=3.-r-f,  and 
^-r-|=J,  or  f  pounds,  the  quotient  being  the  divisor  inverted.  Again, 
if  $1  will  buy  |  pounds,  $|  will  buy  f  of  f  pounds ;  and  |  of  i=f, 
or  1 1  pounds,  the  same  as  before. 

Remarks. — 2.  In  this  analysis,  it  will  be  seen,  by  inspection,  that 
the  numerator  oi  each  fraction  is  also  multiplied  into  the  denomi- 
nator of  the  other.  Both  of  these  solutions,  therefore,  bring  the 
same  combinations  of  terms,  and  the  same  result,  as  inverting  the  di- 
visor, and  multiplying  the  dividend  by  it.     (Art.  166.)    Hence,  the 

EuLE. — Reduce  tlie  fractions  to  a  com.  denominator,  and 
divide  the  numerator  of  the  divideyid  hy  that  of  the  divisor. 
Or,  multiply  the  dividend  by  the  divisor  inverted. 

Notes. — i.  The  first  method  is  based  upon  the  principle  that 
lumerators  of  fractions  having  a  com.  denom.  are  like  numbers. 

The  second  is  evident  from  the  fact  that  it  brings  the  same  com- 
binations as  reducing  the  fractions  to  a  common  denominator. 

The  divisor  is  inverted  for  convenience  in  multiplying, 

2.  After  the  divisor  is  inverted,  the  common  factors  should  be 
cnnceiled,  before  the  multiplication  is  performed. 

3.  Mixed  numbers  should  be  reduced  to  improper  fractions,  com- 
pound and  complex  fractions  to  simj)le  ones. 

4.  Those  expressions  which  have  fractional  denominators,  are  tq- 
redncedtosimple  fractions  hy  the  above  rule.  Ex.14,  (^^t  141,  Rem.) 

170.  How  when  they  have  not?  Note.  Upon  what  principle  is  the  first  method 
based  ?  The  eecond  ?  Show  this  coincidence.  What  is  done  with  mixed  nana 
bers,  compound,  and  complex  fractions. 


lO. 

TTn;-^ 

A'tt. 

11. 

m~ 

■ffS- 

12. 

f-ii- 

■tWj- 

13- 

t¥A 

-T¥/y 

130  DIVISION     OF     FRACTIOKS- 

Perform  the  following  divisions : 

.,4_i_3  »tIS_-_9 

3-  TT  •   TT-  /•    -3T  •   ^y 

4-  A- A-  8.  i-i^U- 

14.  Eeduce  -^J  to  a  simple  fraction. 

Solution. — The  gWen  expression  is  equivalent  to  27-^ -f- to \,  and  is 
reduced  to  a  simple  fraction  by  performing  the  division  indicated. 
A71S.  I  or  2|.     (Art.  141,  Rem.) 

A  I  T  C  ^ 

15.  Eeduce-^  17.  Reduce-^ 

T  T -  T  '7o ^ 

16.  Rednce  — f-  18.  Reduce  -^^ 

19.  Divide  777-1^  dollars  by  129!^  dollars. 

20.  How  many  rods  in  23 20 J  feet,  at  16^  feet  to  a  rod  ? 

21.  How  many  times  30J  sq.  yards  in  320yi3-  sq.  yards? 

22.  What  is  the  quotient  of  f  of  J  of  4^  divided  by  f  of  |? 

2 


23  2       5  o 

Solution. — -  of  -  of  4A  divided  by  -  of  „  =      ^ 

3         4  7^4 


-    X 


X  1-  X  ^  X  -  =  -^  —  124.  Ans. 


S      4      2      2      5         5  ^  ^17 

^^  When  the  perpendicular  form  is  adopted,  the      5 


dicisor  must  b3  inverted,  before  its  terms  are  arranged.     5 1^3  —  ^  2-| 
23.  Di^ddefoff  of|by|ofi-. 
24.  Divide  ^  of  J  of  f  by  i  of  J. 

25.  Divide  f  of  f  of  4I  by  -|  of  J  of  4^. 

.1  g 

26.  What  is  the  quotient  of  |  divided  by  -^  ? 

3  3? 

Analysts.— The  dividend  |  rrr  i  x  4 ;  the  divisor  -_-=:-  X  -^. 
f  3i      I       16 

I     S     I     lis      I 
Invertingthe  divisor,  etc.,  ^^  ^  x  ^  X  -  X  y =—   ^^^^  ^^^^  ^  ^^ 

27.  What  is  quotient  of -^  divided  by  -^? 


DITISION"     OF     FRACTION'S,  131 

171.  The  preceding  principles  may  be  reduced  to  the 
following 

GENERAL    RULE. 

Reduce  whole  arid  mixed  numbers  to  improper  fractions, 
compound  and  complex  fractions  to  simple  ones,  and  mul- 
tiply the  dividend  hy  the  divisor  inverted. 

Note. — After  the  divisor  is  inverted,  the  common  factors  should 
be  cancelled. 

EXAMPLES. 

1.  If  a  young  man  spends  $2|  a  month  for  tobacco,  in 
what  time  will  he  spend  813  J  ?     (Art.  48,  Note  3.) 

2.  If  a  family  use  5-J:  pounds  of  butter  a  week,  how  long 
will  45  ^-  pounds  last  them  ? 

3.  If  $1  will  buy  I  yard  of  gingham,  how  much  will 

4.  How  many  tons  of  coal,  at  $7}  a  ton,  can  be  bought 
for  $1251? 

5.  How  many  times  will  a  keg  containing  13I  gallons 
of  molasses  fill  a  measure  that  holds  |^  of  a  gallon  ? 

6.  What  is  the  quotient  of  2^^^  divided  by  8f  ? 

7.  What  is  the  quotient  of  f|  divided  by  If? 

8.  What  is  the  quotient  of  \l%  divided  by  J|  ? 

9.  What  is  the  quotient  of  45  J  divided  by  25  J  ? 

10.  How  much  tea,  at  $i|  a  pound,  can  be  bought 

for$75l? 

11.  How  many  acres  can  be  sowed  with  57!  bushels  of 

oats,  allowing  if  bushel  to  an  acre  ? 

12.  A  man  having  57I  acres  of  land,  wished  to  fence  it 
into  lots  of  5f  acres :  how^  many  lots  could  he  make  ? 

13.  How  many  yards  of  cloth,  at  $6f,  can  you  buy  for 
$268f ? 

14.  What  is  the  quotient  of  |  of  f  of  Jf  H-f  of  |  of  2 J  ? 

15.  What  is  the  quotient  of  f  of  J  of  8f -^4f  |  ? 


[71.  What  is  the  general  rnle  for  dividing:  fractious  f 


132  DIVISION     OF     FKACTIOI^S. 


QUESTIONS     FOR     REVIEW. 

1.  A  book-keeper  adding  a  column  of  figures,  made  the 
T3sulfc  $563!;  proving  his  work,  he  found  the  true  amount 
to  be  I607J:  how  much  was  the  error? 

2.  A  merchant  paid  two  bills,  one  I278I,  the  other 
$34of,  calling  the  amount  I638J :  what  should  he  have 
paid  ?    What  the  error  ? 

3.  What  is  the  sum  of  35I  and  23I  minus  8|? 

4.  A  speculator  bought  two  lots  of  land,  one  containing 
47x77  acres,  the  other  6:^f  acres:  after  selling  78^  acres, 
how  many  had  he  left  ? 

5.  What  is  the  sum  of  ^-J  plus  \  ? 

6.  What  is  the  sum  --  plus  -    plus  — ,-  ? 

4  '        12  ^       5j 

7.  A  man  owning  f|^  of  a  ship  worth  I48064,  sold  J  of 
his  share.  What  part  of  the  ship  did  he  sell ;  what  part 
does  he  still  own,  and  what  is  it  worth  ? 

8.  A  farmer  owning  75!  acres,  sold  31 J  acres,  and  after- 
ward bought  42  J  acres :  how  many  acres  did  he  then  have  ? 

3  ^5 

9.  What  is  the  difference  between  |  and  ~  ? 

6  k1 

10.  What  is  the  difference  between  -^  and  ~  ? 

si  6 

11.  What  is  the  difference  between  --—  and  — J? 

H  12^ 

12.  If  it  requires  i  J  bushels  of  wheat  to  sow  an  acre, 
how  many  bushels  will  be  required  to  sow  28|  acres? 

13.  How  many  feet  in  148I-  rods,  allowing  16 J  feet  to 
a  rod? 

14.  If  a  pedestrian  can  walk  45^  miles  in  i  day,  how 
far  can  he  walk  in  18J  days  ? 

15.  What  will  37 J  barrels  of  apples  come  to,  at  $2|  per 
barrel  ? 


QUESTIONS     FOR     REYIEW.  133 

1 6.  The  sum  of  two  numbers  is  68 |f,  and  the  difference 
between  them  is  i3f :  what  are  the  numbers  ? 

17.  What  is  the  product  of  4  i^to  ^f  ? 

18.  What  is  the  product  of  ^  into  ^^  ? 

6f  12 

19.  What  is  the  product  of — f  into  -|? 

1 2 1-  2  J 

20.  How  many  days'  work  will  100  men  perform  in  || 
of  a  day  ? 

21.  A  man  owning  y\  of  a  section  of  land,  sold  -}  of  his 
share  for  $i2f :  what  is  the  whole  section  worth,  at  that 
rate? 

22.  How  many  times  is  -^jj  of  f  of  5 J  contained  in  23I ? 

23.  Divide  f  of  i8|  by  f  of  f ^  of  f  of  3i|. 

24.  If  a  gang  of  hands  can  do  ^  of  a  job  in  5-^  days, 
I  ^hat  part  of  it  can  they  do  in  i  day  ? 

A     25.  If  f  of  a  yard  of  satin  will  make  i  vest,  how  many 
vests  can  be  made  from  31^  yards  ? 

26.  How  many  oil-cans,  each  containing  if  gallon,  can 
be  filled  from  a  tank  of  6i|  gallons  ? 

27.  If  a  man  walks  3 J  miles  an  hour,  how  long  w^ill  it 
take  him  to  walk  45 ^^  miles  ? 

28.  By  what  must  JJ  be  multiplied  to  produce  15!? 

29.  How  many  bushels  of  apples,  at  f  of  a  dol.,  are 
required  to  pay  for  6  pair  of  boots,  at  $6 J  ? 

30.  A  farmer  sold  330^^  pounds  of  maple  sugar,  at  i6| 
cents  a  pound,  and  took  his  pay  in  muslin,  at  22 J  cents  a 
yard:  how  many  yards  did  he  receive? 

31.  Divide  the  quotient  of  12^  divided  by  3I  by  the 
quotient  of  6J-f-  3I  ? 

32.  What  is  the  quotient  of  --  —  -4- ? 

T  TT 

7,\  12 

33.  What  is  the  quotient  of  12^  times  -?■  4-  ~j? 


134  FRAGTIOKAL     RELATION 


FRACTIONAL    RELATION    OF    NUMBERS. 

172.  That  Numbers  may  be  compared  with  each  other  frac^ 
iionahy,  they  must  be  so  far  of  the  same  nature  that  one  may  prop- 
erly be  said  to  be  a  part  of  the  other.  Thus,  an  inch  may  be  com- 
pared with  2ifoot ;  for  one  is  a  twelfth  part  of  the  other.  But  it  can- 
not be  said  that  9,  foot  is  any  part  of  an  hour  ;  therefore  the  former 
cannot  be  compared  with  the  latter. 

173.  To  find  what  part  one  number  is  of  another. 

1.  What  part  of  4  is  i  ? 

Analysis. — If  4  is  divided  into  4  equal  parts,  one  of  those  parts  is 
called  I  fourth.     Therefore,  i  is  -^  part  of  4. 

2.  What  part  of  6  is  4  ? 

Analysis. — i  is  \  of  6,  and  4  is  4  times  \,  or  %  of  6.  But  |=| 
(Art.  146) ;  therefore,  4  is  §  of  6.     Hence,  the 

Rule. — Make  the  number  denoting  the  part  the  numera- 
tor, and  that  ivith  which  it  is  compared  the  denominator. 

Note. —  i.  This  rule  embraces /<9wr  classes  of  questions : 
ist.  What  part  one  wJiole  number  is  of  another, 
2d.  What  part  a  fraction  is  of  a  whole  number. 
3d.  What  part  a  whole  number  is  of  a  fraction. 
4th.  What  part  one  fraction  is  of  another. 

2.  When  complex  fractions  occur,  they  should  be  reduced  to 
wimple  ones,  and  all  answers  to  the  lowest  terms.     (Art.  146.) 

3.  What  part  of  75  is  15  ?     Of  84  is  30  ? 

4.  Of  91  is  63?      6.  Of    81  is  18?      8.  Of  256  is    72? 

5.  Of  48  is  72  ?    .   7.  Of  100  is  75  ?      9.  Of  375  is  425  ? 

10.  What  part  of  i  week  is  5  days? 

11.  A  man  gave  a  bushel  of  chestnuts  to  17  boys:  what 
part  did  5  boys  receive  ? 

12.  At  $13  a  ton,  how  much  coal  can  be  bought  for  $10  ? 

13.  A  father  is  51  years  old,  and  his  son's  age  is  17: 
what  part  of  the  father's  age  is  the  son's  ? 


OF     K^  UMBERS.  135 

14.  If  8  pears  cost  35  cents,  what  will  5  pears  cost  ? 

Analysis. — 5  pears  are  f  of  8  pears ;  hence,  if  8  pears  cost  35 
cents,  5  pears  will  cost  f  of  35  cents.  Now  ^  of  35  cents  is  4I  cents, 
And  5  eighths  are  5  times  4^  cents,  which  are  21^  cents. 

15.  If  5  bar.  of  flour  cost  $45,  what  w^ill  28  bar.  cost? 

16.  If  50  yds.  of  cloth  cost  $175,  what  will  17  cost? 

17.  If  25  bu.  of  apples  cost  $30,  what  will  no  bu.  cost? 
t8.  What  part  of  5  is  f  ? 

Analysis. — Making  the  fraction  which  denotes  the  part  the  nu- 
merator, and  the  whole  number  the  denominator,  we  have  a  fraction 
to  be  divided  by  a  whole  number.  For,  all  denominators  may  be  con- 
sidered as  divisors.    Thus,  ^-i-s=-^o,  Ans.    (Art.  142.) 

19.  What  part  of  25  is  J?        21.  What  part  of  30  is  -^^ ? 

20.  What  part  of  35  is  -^^^-  ?      22.  What  part  of  40  is  ^  ? 

23.  If  5  acres  of  land  cost  $100,  what  will  J  acre  cost  ? 

Analysis. — i  acre  is  ^  of  5  acres,  and  f  of  an  acre  is  |  of  i,  or  -^^ 
of  5  acres.  Hence,  ^  of  an  acre  will  cost  2^0  of  $100.  NoWa^,-  of  $100 
is  $5  ;  and  3  twentieths  are  3  times  5  or  $15. 

24.  When  coal  is  $95  for  15  tons,  what  will  f  ton  cost? 

25.  If  19  yards  of  silk  cost  $60,  what  will  |  yard  cost? 

26.  What  part  of  f  is  2  ? 

Analysis. — Making  the  wJiole  number  which  denotes  the  part,  ths 
numerator,  and  the  fraction  the  denominator,  we  have  a  whole  num- 
ber to  be  divided  by  a  fraction.     Thus,  2^|— 2  x  §— \^,  Ans. 

27.  What  part  of  f  is  8  ?  29.  What  part  of  f  is  11? 

28.  What  part  of  |  is  12  ?  30.  What  part  of  ^V  is  20  ? 

31.  What  part  off  is  f? 

Analysis. — Making  the  fraction  denoting  the  part  the  numerator, 
and  the  other  the  denominator,  we  have  a  fraction  to  be  divided  by 
a  fraction.     Thus,  f^|=|  x  f =f,  ^/^.s.     (Art.  170.) 

32.  What  part  of  |  is  f  ?  34.  Wliat  part  of  |f  is  ^^  ? 
SS.  What  part  of  f|  is  fj  ?  35.  What  part  of  i|  is  f |  ? 
$6.  6J  is  what  part  of  25  ? 

Analysis. — Reducing  the  mixed  number  to  an  improper  fraction, 
we  have  6^=*/,  and  \^-T-25  =  i-,  Ans. 


136  FRACTIONAL     RELATIOi^. 

37.  What  part  of  100  is  12^?  40.  Of  100  is  62J  ? 

38.  What  part  of  100  is  $si^  4i-  C)f  100  is  iSJ? 

39.  What  part  of  100  is  i6f  ?  42.  Of  100  is  87^? 

43.  12 J  is  what  part  of  18J  ? 

Analysis. — Reducing  the  mixed  numbers  to  improper  fractions 
we  have,  12^=^",  and  i8|=^4^.     Now  ^/-f-i/=f,  Ans. 

44.  What  part  of  62^  is  i8|  ?  45.  Of  874  is  31^  ? 

46.  At  til  ^  pound,  how  much  tea  will  $f  buy  ? 

47.  At  $JJ  per  foot,  liow  niany  feet  of  land  can  be 
bought  for  $ii  ? 

48.  A  lad  spent  iSJ  cents  for  candy,  which  was  62-J 
cents  a  pound :  how  much  did  he  buy  ? 

49.  A  can  do  a  certain  job  in  8  days,  and  B  in  6  days  * 
what  part  will  both  do  in  i  day  ? 

50.  What  part  of  4  times  20  is  9  times  16  ? 

51.  What  part  of  75  x  18  is  Io5-^25  ? 

52.  What  part  of  {6S  —  24)  x  14  is  168  -^  12  ? 

174.  To  find  a  Number,  a  Fractional  Part  of  it  being  giver*. 

Ex.  I.  9  is  I  of  what  number  ? 

Analysis. — Since  9  is  i  third,  3  thirds  or  the  whole  number  must 
be  3  times  9  or  27.     Therefore,  g  is  a  third  of  27. 

Or,  thus :  q  is  ^  of  3  times  9,  and  3  x  9=27.     Therefore,  etc. 

2.  21  is  J  of  what  number? 

Analysis. — Since  f  of   a  certain  number  is  operation. 

21  units,  I  or  the  whole  number  must  be  as  many      2l-r-|:— 21  X^ 
units  as  |  are  contained  times  in  21 ;  and  2i-;-|=:      21x^=2  8. 
21  X  ^=28,  the  answer  required.     For,  a  whole 
number  is  divided  by  a  fraction  by  multiplying  the  former  by  the 
latter  inverted.     (Art.  168,  a.) 

Or,  thus :  Since  21  is  |  of  a  certain  number,  i  fourth  of  it  is  i  third 
of  21,  or  7.  Now  as  7  is  I  fourth  of  the  number,  4  fourths  must  be 
4  times  7  or  28,  the  same  as  before.     Hence,  the 

Rule. — Divide  the  number  denoting  the  part  hy  the 
fraction. 

Or,  Find  one  part  as  indicated  hy  the  numerator  of  th^ 
fraction^  and  multiply  this  hy  the  denominator. 


OF     LUMBERS.  137 

Note. — The  learner  Bhould  observe  the  difference  between  finding 
^  of  a  number,  when  f  or  the  whole  number  is  given,  and  when  only 
f  or  a  part  of  it  is  given.  In  the  former,  we  divide  by  the  denomi- 
nator of  the  fraction ;  in  the  latter,  by  the  numerator,  as  in  the 
second  analysis.  If  he  is  at  a  loss  which  to  take  for  the  divisor,  let 
Mm  substitute  the  word  parts  for  the  denominator. 

3.  56  is  f  of  what  ?  7.  436  is  f  of  what  ? 

4.  68  is  J  of  what?  8.  456  is  f  of  what? 

5.  85  is  f  of  what  ?  9.  685  is  -^  of  what  ? 

6.  1 15  is  f  of  what  ?  10.  999  is  y^  ^f  what  ? 

1 1.  A  market  man  being  asked  how  many  eggs  he  had, 
replied  that  126  was  equal  to  -^  of  them :  how  many  had  he  ^ 

12.  If  y\  of  a  ship  is  worth  $8280,  what  is  the  whole 
worth  ? 

13.  A  commander  lost  f  of  his  forces  in  a  battle,  and 
had  9500  men  left:  how  many  had  he  at  first? 

14.  fj  is  :!  of  what  number? 

Analysis.— f^  is  i  of  4  times  f^' ;  and  4  times  fr=2|. 
i5«  fi  is  y  of  what  number  ? 

Analysis. — Since  ii^?  of  a  certain  number,  \  of  that  number 
must  be  \  of  H,  and  ^  of  fl^/s,  or  \.  Now  if  \  of  the  number=:;^, 
^  must  equal  7  times  \=\,  or  i|,  Ans. 

7,  21     %     7 
Or,  dividing  |i  by  f  we  have       --  x  .  =-,  or  i J,  Ans. 

4,   2J.O.        A        4 

16.  If  is  I  of  what  number  ? 
17-  If  is  f  of  what  number? 
18.  18J  IS  f  of  what  number? 

Analysis.— 1 8|=^4^    Since  \^  =  1,  1  =  '^^-,  and  ^=^ja  or  30,  Ans^ 
Or,  \'^-H-^=^/ X  «=fi^,i=3o,  the  same  as  before. 

^9'  372  is  f  of  what  number? 

20.  66f  is  f  of  what  number  ? 

21.  48  is  f  of  f  of  what  number  ? 

Analysis. — f  of  ^=^.  The  question  now  is,  48  is  f  of  what 
number  ?    Ans.  48  x  f ,  or  108. 


133  FRA.CTIOKAL     RELATIONS. 

2  2.  1 1 2  is  I  of  f  of  what  number  ? 

23.  In  i  of  120  how  many  times  15  ? 

Analysis. — h  of  120  ie  13^;  and  I  are  7  times  13^  or  93^.  I^ow 
93i-i5=^l^-i5=\¥  or  6|,  ^tis. 

24.  How  many  yards  of  brocatelle,  at  I9  a  yard,  can  be 
bought  for  -J  of  $100  ? 

25.  A  man  paid  ^  of  $280  for  84  arm-chairs :  what  was 
tliat  apiece  ? 

26.  90  is  f  of  how  many  times  17  ? 

Analysis. — As  90  is  &  of  a  certain  number,  i  is  ^  of  90,  which  is 
15  ;  and  t  are  7  times  15  or  105.  Now  17  is  in  105,  6-^-  times. 
Therefore,  etc. 

27.  125  is  f  of  how  many  times  20  ? 

28.  A  man  paid  60  cents  for  his  lunch,  which  was  ^  of 
his  money,  and  spent  the  remainder  for  cigars,  which  were 
5  cents  each :  how  much  money  had  he ;  and  how  many 
cigars  did  he  buy  ? 

29.  ^  of  no  is  f  of  what  number  ? 

Analysis. —  i^n  of  no  is  11,  and  1%,  9  times  11  or  99.  Now,  since 
99  is  t  of  a  number,  y  of  it  must  be  ^  of  99,  which  is  11,  and  ^  must 
be  7  times  11  or  77.     Therefore,  etc. 

30.  1^  of  126  is  J  of  what  number  ? 

31.  f  of  90  is  I  of  how  many  times  11  ? 

Analysis. — f  of  90  is  50.  Now  as  50  is  f  of  a  number,  ^r  is  i  of 
50  or  10;  I  is  8  times  10  or  80.  Finally,  11  is  contained  in  80,  7i^ 
times.     Therefore  ^  of  90  is  §  of  7 1\  times  11. 

32.  -I  of  96  is  3^  of  how  many  times  20  ? 

^^,  f  of  120  is  1^  of  how  many  sevenths  of  56  ? 

Analysis. — g  of  120  is  100.  If  100  is  |  of  a  number,  ^  is  \  of  the 
number;  now  \  of  100  is  25,  and  |  is  9  times  25  or  225.  Finally,  f 
of  56  is  8,  and  8  is  contained  in  225,  28^  times.  Therefore,  ^  of  120 
is  f  of  28^  times  |  of  56. 

34.  f  of  35  is  I  of  liow  many  tenths  of  120? 


"DECIMAL   FRACTIONS. 

175.  Decimal  Fractions  are  those  in  which  the 
unit  is  divided  into  tenths,  hundredths,  thousandths,  etc. 
They  arise  from  continued  divisions  by  lo. 

If  a  unit  is  divided  into  ten  equal  parts,  the  parts  are 
called  tenths.  JSTow,  if  one  of  these  tenths  is  subdivided 
into  ten  other  equal  parts,  each  of  these  parts  will  be  one- 
tentli  of  a  tenth,  or  a  hundredth.  Thus,  -f^—  lo  or  ^  of 
^—^^.  Again,  if  one  of  these  hundredths  is  subdivided 
into  ten  equal  parts,  each  of  these  parts  will  be  one-tenth  of 
a  hundredth,  or  a  thousandtli.     Thus,  ttJ-jj-t-  ^o—-^-}^^,  etc. 

NOTATION    OF    DECIMALS. 

176.  If  we  multiply  the  unit  i  by  lo  continually,  it 
produces  a  series  of  whole  numbers  which  increase  regu- 
larly by  the  scale  of  lo ;  as, 

I,     lo,     loo,    looo,    loooo,    looooo,    loooooo,    etc. 

Now  if  we  divide  the  highest  term  in  this  series  by  lo 
continually,  the  several  quotients  will  form  an  inverted 
series,  which  decreases  regularly  by  ten,  and  extends  from 
the  highest  term  to  i,  and  from  i  to  xV?  ttottj  tx^>  ^^^  so 
on,  indefinitely ;  as, 

looooo,   loooo,  looo,  loo,  10,  I,  ^,  yj^,  ToW^   ^tc. 

177.  By  inspecting  this  series,  the  learner  wiU  perceive 
that  ^'^  fractions  thus  obtained,  regularly  decrease  toward 
the  right  by  the  scale  of  lo. 

If  we  apply  to  this  class  of  fractions  the  great  law  of 
Arabic  Notation,  which  assigns  different  values  to  figures, 

175.  What  are  decimal  fractions?  How  do  they  arise?  Explain  this  upon 
Uie  blackboard.    177.  By  what  law  do  decimals  decrease  ? 


140  DECIMAL     FRACTIOI^S. 

according  to  the  place  they  occupy,  it  follows  that  a  figure 
standing  in  the  first  place  on  the  rigid  of  units,  denotes 
tenths,  or  i  tenth  as  much  as  when  it  stands  in  units' 
place ;  when  standing  in  the  second  place,  it  denotes  hun- 
dredths, or  I  tenth  as  much  as  in  the  first  place ;  when 
standing  in  the  third  place,  it  denotes  thousandths,  etc., 
each  succeeding  order  below  units  being  one  tenth  the 
value  of  the  preceding.     Hence, 

178.  Fractions  which  decrease  by  the  scale  of  ten,  may 
be  expressed  like  whole  numbers ;  the  value  of  each  figure 
in  the  decreasing  scale  being  determined  by  the  place  it 
occupies  on  the  right  of  units.  Thus,  3  and  5  tenths  may 
be  expressed  by  3.5 ;  3  and  5  hundredths  by  3.05 ;  3  and 
5  thousandths  by  3.005,  etc. 

178,  <i'  Decimals  are  distinguished  from  whole  numbers 
by  a  decimal  'point. 

Notes. — i.  The  decimal  point  commonly  used  (.),  is  a  period. 
2.  This  class  of  fractions  is  called  decimals,  from  the  Latin  decern, 
ten,  which  indicates  both  their  origin  and  the  scale  of  decrease. 

179.  The  Deno7ninator  of  a  decimal  fraction  is 
always  10,  100,  1000,  etc.;  or  i  with  as  many  ciphers 
annexed  to  it  as  there  are  decimal  places  in  the  given 
numerator.     Conversely, 

Tlie  Numerator  oi  a  decimal  fraction,  when  written 
alone,  contains  as  many  figures  as  there  are  ciphers  in  the 
denominator.  Thus  -^,  -j^,  rwuuy  expressed  decimally, 
are  .5,  .05,  .005,  etc.  If  the  ciphers  in  .05  and  .005  are 
omitted,  each  becomes  5 -tenths. 


What  place  do  tenths  occupy  ?  Hundredths,  thousandths,  etc.  ?  178.  How  is 
the  value  of  decimal  figures  determined  ?  178,  a.  How  are  decimals  diBtinguished 
from  whole  numbers?  JVote.  Wliat  is  the  decimal  point?  From  what  does  this 
tlass  of  fractions  receive  its  name?  179.  What  is  the  denominator?  How  do 
the  number  of  ciphers  in  the  denominator  and  the  decimal  places  in  the  numera' 
tor  compare  ? 


DECIMAL     FRACTIONS.  141 

180.  The  different  orders  of  decimals,  and  their  relative 
position,  may  be  seen  from  the  following 

TABLE. 


■5  a5 


r2  TS  '^ 

£     o     « 


)^    K    H    ^    W    H    i^ 
8425672 


.s 

^ 

S 

rC3 

;=H 

a 

1 

1 

0 

0 

1 

^3 

OB 

.2 

1 

•73 

a 

d 

S 

g 

a 

a 

S 

3    ^ 


Integers.  ^  Decimals. 

Notes. — i.  A  Decimal  and  an  Integer  written  together,  are  called 
R.  Mixed  Number ;  as  35.263,  etc.     (Art.  loi,  Def.  12.) 

2.  A  Decimal  and  a  Common  Fraction  written  together,  are  called 
ft  Mixed  Fraction  ;  as  .6^,  .33^. 

181.  Since  the  orders  of  decimals  decrease  from  left  to 
right  by  the  scale  of  i  o,  it  follows : 

First.  Prefixing  a  cipher  to  a  decimal  diminishes  its  value 
10  times,  or  divides  it  by  lo.     Thus  .7— -j-'^;  .07=^^^; 

•oo7  =  t(fW 

Annexing  a  cipher  to  a  decimal  does  not  alter  its  value. 
Thus  .7,  .70,  .700  are  respectively  equal  to  y^^,  y^^*^,  y^xA? 

or  A- 

These  effects  are  the  reverse  of  those  produced  by  annexing  and 
prefixing  ciphers  to  whole  numbers.     (Arts.  57,  79.) 

Second.  Each  removal  of  the  decimal  point  one  figure 
to  the  righty  increases  the  number  10  times,  or  multiplies 
it  by  10.  Each  removal  of  the  decimal  point  one  figure 
to  the  left f  diminishes  the  number  10  times,  or  divides  it 
by  10.   Hence. 

180.  Name  the  orders  of  decimals  toward  the  right.  Name  the  orders  of 
integers  toward  the  left.  i8i.  How  is  a  decimal  figure  affected  by  moving  it  one 
place  to  the  right?  How,  if  a  cipher  is  prefixed  to  it?  How  if  ciphers  are 
annexed  ? 


142  DECIMAL     FEACTI0:N"8. 

182.  To  write  Dechnals, 

EuLE. —  Write  the  figures  of  the  numerator  in  their  order^ 
assigning  to  each  its  proper  place  beloiv  units,  and  prefix  to 
them  the  decimal  point. 

If  the  numerator  has  not  as  many  figures  as  required ^ 
siipply  the  deficiency  by  prefixing  ciphers. 


•ite 

)  the  following  fractions 

decimally : 

I. 

A-             6.  1^. 

II-  93tttW 

2. 

AV               7.  t'A- 

12.       7T4fT)- 

3- 

AV               8.     4iV. 

13-   lOn/^Ji 

4- 

t'A.                       9-    2lTfcT. 

14-  46twoif- 

5- 

T*T7.                   lO-    84tU. 

15-  So^Mi 

17.  Write  6  hundredths;  6^  thousandths;  109  ten 
thousandths. 

18.  Write  305  thousandths;  21  hundred-thousandths: 
95  millionths. 

19.  Write  4  thousandths ;  108  ten-thousandths;  46  hun- 
dreths;  65  millionths;  1045  ten-millionths. 

20.  Write  sixty-nine  and  four  thousandths;  ten  and 
seventy-five  ten -thousands;  160  and  6  millionths. 

21.  Write  53  ten-thonsandths ;  67,  and  28  hundred- 
thousandths;  352  ten-millionths. 

183.  To  read  Decimals  expressed  by  Figures. 

Rule. — Read  the  decimals  as  whole  numbers,  and  apply 
to  them  the  name  of  the  loivest  order. 

Note. — In  case  of  mixed  numbers,  read  the  integral  part  as  it"  it 
stood  alone,  then  the  decimal. 

Or,  pronounce  the  word  decimal,  then  read  the  decimal  figures  aa 
if  they  were  whole  numbers. 

182.  WTiat  is  the  rale  for  writing  decimals?  Note.  If  the  numerator  has  not 
as  many  figures  as  there  are  ciphers  in  the  denominator,  what  is  to  be  done? 
183.  How  arc  decimals  read  ?    Note.  In  case  of  a  mixed  number,  how  ? 


DECIMAL     FRACTIONS.  143 

Or,  having  pronounced  the  word  decimal,  repeat  the  names  of  the 
decimal  figures  in  their  order.  Thus,  275.468  is  read,  "  275  and  468 
thousandths;"  or  "  275,  decimalfour  hundred  and  sixty-eight ; "  or 
"  275,  decimal  four,  six,  eight." 

I.  Explain  the  decimal  .05. 

Analysis. — Since  the  5  stands  in  the  second  place  on  the  right  of 
the  decimal  point,  it  is  equivalent  to  j^o,  and  denotes  5  hundredths 
of  one,  or  5  such  parts  as  would  be  obtained  by  dividing  a  unit  into 
100  equal  parts. 

Bead  the  following  examples : 

(I.)  (2.)  .  (3.) 

.^6  2.751  32.862 

•479  4.8465  40.0752 

.0652  7-25025  57.00624 

.00316  8.400452  81.20701 

183  ff"  Decimals  differ  from  common  fractions  in  three 
respects,  viz.  :  in  their  origin,  their  notation,  and  their  limitation. 

1st.  Common  fractions  arise  from  dividing  a  unit  into  any  number 
of  equal  parts,  and  may  have  any  number  for  a  denominator. 

Decimals  arise  from  dividing  a  unit  into  ten,  one  hundred,  one 
thousand,  etc.,  equal  parts  ;  consequently,  the  denominator  is  always 
10,  or  some  power  of  10.     (Arts.  55,  n.  179.) 

2d.  In  the  notation  of  common  fractions,  both  the  numerator  and 
denominator  are  written  in  full. 

In  decimals,  the  numerator  only  is  written ;  the  denominator  is 
understood. 

3d.  Common  fractions  are  universal  in  their  application,  embracing 
all  classes  of  fractional  quantities  from  a  unit  to  an  infinitesimal. 

Decimals  are  limited  to  that  particular  class  of  fractional  quanti- 
ties whose  orders  regularly  decrease  in  value  from  left  to  right,  by 
the  scale  of  10. 

Note. — The  question  is  often  asked  whether  the  expressions  1%, 
T^o>  tthtut  etc.,  are  common  or  decimal  fractions. 

All  fractions  whose  denominator  is  vyritten  under  the  numerator, 
fulfil  the  conditions  of  common  fractions,  and  may  be  treated  as  such. 
But  fractions  which  arise  from  dividing  a  unit  into  10, 100,  1000,  etc., 
equal  parts,  ansicer  to  tJie  definition  of  decimals,  whether  the  denom- 
inator is  expressed,  or  understood. 

i33,  a.  How  do  decimals  differ  from  coiuuiou  fraction..-^  ? 


14:4  DECIMAL    FRACTIONS. 

REDUCTION   OF   DECIMALS. 

184.  To  Reduce  Decimals  to  a  Common  Denom^inator, 

1.  Reduce  .06,  2.3,  and  .007  to  a  common  denominator. 
Analysis  — Decimals  containing  the  same  number       .06  =  0.060 

of  figures,  have  a  common  denominator.     (Art.  179.)        «  •3  =  2  -200 

By  annexing  ciphers,  the  number  of  decimal   fig-  

ares  in  each  may  be  made  the  same,  without  altering     *      '        *      ' 
their  value.     (Art.  181.)    Hence,  the 

Rule. — Make  the  number  of  decimal  figures  the  same  in 
eachf  hy  annexing  ciphers.     (Art.  181.) 

2.  Reduce  .48  and   .0003   to   a  common  denominator. 

3.  Reduce  2  to  tenths ;  3  to  hundredths ;  and  .5  to 
thousandths.  Ans.  2.0  or  f^;  3.00  or  f^f;  .500. 

185.  To  Reduce  Decimals  to  Common  Fractions. 

1.  Reduce  .42  to  a  common  fraction. 

Analysis, — The  denominator  of  a  decimal  is  i,  with  as  many 
ciphers  annexed  as  there  are  figures  in  its  numerator;  therefore  the 
denominator  of  .42  is  100.     (Art.  179.)    Ans,  .42=-j^o%.     Hence,  the 

Rule. — Erase  the  demmal  pointy  and  place  the  denomi- 
nator under  the  numerator.    (Art.  179.) 

2.  Reduce  .65  to  a  common  fraction;  then  to  its  lowest 
terms.  Ans.  .65  =  A^ir^  ^^^  Twu=ii- 

Reduce  the  following  decimals  to  common  fractions  : 

3.  .128  7.  .05  II.  .0007  15.  .200684 

4.  .256  8.  .003  12.  .04056        16.  .0000008 

5.  .375  9.  .0008  13.  .00364        17.  .12400625 

6.  .863  10.  .0605  14.  .00005        18.  .24801264 

186.  To  Reduce  Com^moti  Fractions  to  Decimals. 

I.  Reduce  |  to  a  decimal  fraction. 

Analysis — |  equals  i  of  3.  Since  3  cannot  be  divided  operation. 
fcy  8 ;  we  annex  a  cipher  to  reduce  it  to  tenths.  Now  \  8)3.000 
of  30  tenths  is  3  tenths,  and  6  tenths  over.  6  tenths  AnsT^vfk 
reauced  to  hundredths =60  hundredths,  and  |^  of  60 
hundredths =7  hundredths  and  4  hundredths  over.  4  hundred tha 
reduced  to  thousandths =^0  thousandths,  and  \  of  40  thousandtha 
rr5  thousandtlis.     Therefore,  ?l  =  .375.    Hence,  the 


t 


DECIMAL     FRACTIOi^S  145 

Rule — Annex  ciphers  to  the  numerator  and  divide  by  the 
denominator.  Finally,  point  off  as  many  decimal  figures  in 
the  result  as  there  are  ciphers  amiexed  to  the  numerator. 

Note. — If  the  number  of  figures  in  the  quotient  is  less  than  the 
number  of  ciphers  amiexed  to  the  numerator,  supply  the  deficiency 
by  prefixing  ciphers. 

Demonstration. — A  fraction  indicates  division,  and  its  value  is  the 
numerator  divided  by  the  denominator.  (Arts.  134,  142.)  Now,  an- 
nexing one  cipher  to  the  numerator  multiplies  the  fraction  by  10 ; 
annexing  two  ciphers,  by  100,  etc.  Hence,  dividing  the  numerator 
with  one,  two,  or  more  ciphers  annexed,  gives  a  quotient,  10,  100, 
etc.,  times  too  large.  To  correct  this  error  the  quotient  is  divided 
by  10,  100,  1000,  etc.  But  dividing  by  10,  100,  etc.,  is  the  same  as 
pointing  off  an  equal  number  of  decimal  figures.    (Art.  181 .) 

Reduce  the  following  fractions  to  decimals : 

2.  i  6.  I  10.  ^  14.  ^^ 

3.  f  7.  A  1 1- it  15.7!^ 

4-  J  8.  I  12.  ^V  -16.  ^l-Q 

1 8.  Reduce  f  to  the  form  of  a  decimal. 

Analysis. — Annexing  ciphers  to  the  numerator      operation. 
and  dividing  by  the  denominator,  as  before,  the  quo-    3)2.000 
tient  consists  of  6  repeated  to  infinity,  and  the  re-       "^6666   etc. 
mainder  is  always  2.     Therefore  |  cannot  be  exactly 
expressed  by  decimals. 

19.  Reduce  /y  to  the  form  of  a  decimal. 
Analysis. — Having  obtained  th^ee  quotient  operation. 

figures  135,  the  remainder  is  5,  the  same  as  the    37(5.000000 
original  numerator;   consequently,  by  annex-  .135135?  ^tc. 

ing  ciphers  to  it,  and  continuing  the  division, 
we  obtain  the  same  set  of  figures  as  before,  repeated  to  infinity. 
Thcsrefore  ^  cannot  be  exactly  expressed  by  decimals. 

187.  When  the  numerator,  with  ciphers  annexed,  is 
exactly  divisible  by  the  denominator,  the  decimal  is  called 
a  terminate  decimal. 

.J85.  How  reduce  a  decimal  to  a  common  fraction  ?  186.  How  reduce  a  com- 
mon fraction  to  a  decimal  ?  Note.  If  the  number  of  figujes  in  the  quotient  is  less 
t^an  that  in  the  numerator,  what  le  to  be  done  ?  Explain  the  reason  for  pointing 
off  the  quotient. 


116  ADDITION     OF     DECIMALS. 

When  it  is  not  exactly  divisible^  and  the  same  figure  o 
set  of  figures  continually  recurs  in  the  quotient,  the  deci 
nial  is  called  an  interminate  or  circulating  decimal. 

T\\Q  figure  or  set  of  figures  repeated  is  called  the  repetend 
Thus,  the  decimals  obtained  in  the  last  two  examples  ar< 
interminate,  because  the  division,  if  continued  forever,  wil; 
leave  a  remainder.  The  repetend  of  the  i8th  is  6;  thai 
of  the  19th  is  135. 

Notes. — i.  After  the  quotient  has  been  carried  as  far  as  desirably 
the  sign  ( + )  is  annexed  to  it  to  indicate  there  is  still  a  remainder. 

2.  If  the  remainder  is  such  that  the  next  quotient  jBgure  would  be 
5,  or  more,  the  last  figure  obtained  is  sometimes  increased  by  i,  and 
the  sign  (— )  annexed  to  show  that  the  decimal  is  too  large. 

3.  Again,  the  remainder  is  sometimes  placed  over  the  divisor  and 
annexed  to  the  quotient,  forming  a  mixed  fraction.  (Art.  i8c,  n^ 
Thus  if  I  is  reduced  to  the  decimal  form,  the  result  may  be  expressed 
by  .6666+  ;  by  .6667—  ;  or  by  .6666  .|. 

(For  the  further  consideration  of  Circulating  Decimals,  the  student 
is  referred  to  Higher  Arithmetic.) 

Eeduce  the  following  to  four  decimal  places : 

20.  i  22.     f  24.    -i\  26.     Jf 

21.  f  23.    $  25.     -3-\  27.    4i 

Eeduce  the  following  to  the  decimal  form  : 

28.     75f      30-  26ii|-      32.  465^^      34.  lAo-a^ 
29-  136!      31-  346H      2>Z'  523^3-      35-  956^ 

ADDITION    OF    DECIMALS. 

188.  Since  decimals  increase  and  decrease  regularly  by 
the  scale  of  10,  it  is  plain  they  may  be  added,  subtracted, 
multiplied,  and  divided  like  whole  numbers. 

Or,  they  may  be  reduced  to  a  common  denominator,  then 
be  added,  subtracted,  and  divided  like  Common  Fractions. 
(Arts.  156,  184.) 

187.  When  the  numerator  with  ciphers  annexed  is  exactly  divisible  by  tho 
denominator,  what  is  the  decimal  called  ?  If  not  exactly  divisible,  what  ?  What 
Is  the  figure  or  set  of  figures  repeated  called? 


ADDITION    OF    DECIMALS.  147 

189.  To  find  the  Amount  of  two  or  more  Decimals. 

I.  Add  360.1252,  1. 91,  12.643,  and  152.8413. 
Ai^ALYSis. — Since   units    of  tlie  same  order  or  operation. 

like  number's  only  can  be  added  to  eacli  other,  we  3^^'^ ^5^ 

reduce  tlie  decimals  to  a  common  denominator  by  1.9 100 

annexing  ciphers;    or,   which  is   the   same,  by  1 2.6430 

writing  the  decimals  one  under  another,  so  that  152.8413 

the  decimal  points  shall  be  in  a  perpendicular  ^715.527.5195 
line.  (Arts.  28,  n.  156.)  Beginning  at  the  right, 
we  add  each  column,  and  set  down  the  result  as  in  whole  numbers, 
and  for  the  same  reasons.  (Art.  29,  7i.)  Finally,  we  place  the  deci- 
mal point  in  the  amount  directly  under  those  in  the  numbers  added. 
(Art.  178,  a)    Hence,  the 

Rule. — I.  Write  the  numUrs  so  that  the  decimal  points 
shall  stajid  one  under  another,  with  tenths  under  tenths,  etc. 

II.  Beginning  at  the  right,  add  as  in  whole  numbers,  and 
place  the  decimal poitit  in  the  amount  under  those  in  the 
numbers  added. 

Note.— Placing  tenths  under  tenths,  hundredths  under  hundredths, 
etc.,  in  efiect,  reduces  the  decimals  to  a  com.  denominator;  hence 
th«  ciphers  on  the  right  may  be  omitted.    (Arts.  181,  184.) 


(2.) 

(3.) 

(4.) 

(5.) 

41.3602 

416.378 

36.81045 

4.83907 

4.213 

85.1 

.203 

.293 

61.46 

.4681 

5-3078 

.40 

375.265 

4.3S 

87.69043 

5.1067 

482.2982 

Ans. 

375.2956 

9.25 

3.75039 

6.  What  is  the  sum  of  41.371  +2.29  +  73.4024- 1.729? 

7.  What  is  the  sum  of  823.37 +  7.375 +  61. i +.843? 

8.  What  is  the  sum  of  .3925 +  .64 +  .462 +  .7 +.56781  ? 

9.  What  is  the  sum  of  86  005  +  4.0003  +  2.00007  ? 

10.  What  is  the  sum  of  1.7 13 +  2.30 +  6. 400  + 27.004? 

11.  Add  together  7  tenths;  312  thousandths;  46  hun- 
dredths; 9  tenths;  and  228  ten-thousandths. 

12.  Add   together   23   ten-thousandtks ;    23   hundred- 
thousandths;  23  thousandths;  23  hundredths;  and  23. 

i?9.  How  are  decimals  added  ?    How  point  oflf  tlie  amount  f 


148  SUBTRACTIOK     OF     DECIMALS. 

13.  Add  together  five  hundred  seventy-five  and  seven- 
tenths;  two  hundred  fifty-nine  ten-thousandths;  five- 
miUionths;  three  hundred  twenty  hundred-thousandths. 

14.  A  farmer  gathered  lyf  bushels  of  apples  from  one 
tree ;  8^  bushels  from  another ;  loj  bushels  from  another ; 
and  16  J  bushels  from  another.  Required  the  number  of 
bushels  he  had,  expressed  decimally. 

15.  A  grocer  sold  7^  pounds  of  sugar  to  one  customer; 
11.37  pounds  to  another;  lof  pounds  to  another;  25^ 
pounds  to  another;  and  21.^75  pounds  to  another:  how 
many  pounds  did  he  sell  to  all  ? 

SUBTRACTION   OF   DECIMALS. 

190.  To  find  the  Difference  between  two  Decimals. 

I.  What  is  the  difference  between  2.607  ^^^  -7235  ? 
Analysis. — We  reduce  the  decimals  to  a  common  operation. 

denominator,  by  annexing  ciphers,  or  by  writing  the  2.607 

same  orders  one  under  another.     (Art.  189,  n.)    For,  -7235 

units   of   the  same  order  or  like  numbers  only,  can     Ans.  1.8835 
be  subtracted  one  from  the  other. 

Beginning  at  the  right,  we  perceive  that  5  ten-thousandths  can 
not  be  taken  from  o ;  we  therefore  borrow  ten,  and  then  proceed  in 
all  respects  as  in  whole  numbers,     (Art.  38.)    Hence,  the 

Rule. — I.  Write  tlie  less  number  under  the  greater,  so 
That  the  decimal  points  shall  stand  o?ie  under  the  other, 
with  tenths  tinder  tenths,  etc. 

II.  Bcginniiig  at  the  right  hand,  proceed  as  in  subtract- 
ing luhole  numbers,  and  place  the  decimal  point  in  the 
remainder  under  that  in  the  subtrahend. 

Note. — Writing  the  same  orders  one  under  another,  in  effect 
reduces  the  decimals  to  a. common  denominator.    (Art.  189,  n.) 

(2-)  /        (3-)  (4.)  (5-) 

From    13.051  7*0392  20.41  S5.3004 

rake       5.22  *        .43671  3-0425  67.35246 

190.  How  are  decimals  Bubtracted  ?    How  point  off  the  remainder? 


MULTIPLICATION^    OFDECIMALS.  149 

Perform  the  following  subtractions  : 

6.  13.051  minus  5.22.  12.  10  minus  9.1030245. 

7.  7.0392  minus  0.43671.         13.  100  minus  994503067, 

8.  20.41  minus  3.0425.  14.  i  minus  .123456789. 

9.  85.3004  minus  7.35246.       15.  i  minus  .98764321. 

10.  93.38  minus  14.810034.       16.  o.i  minus  .001. 

11.  3  minus  0.103784.  17.  o.oi  minus  .00001. 

18.  From  100  take  i  thousandth. 

19.  From  45  take  45  ten-thousandths. 

20.  From  I  ten- thousandth  take  2  millionths. 

21.  A  man  having  $673,875,  paid  $230.05  for  a  horse: 
how  much  had  he  left  ? 

22.  A  father  having  504.03  acres  of  land,  gave  100.45 
acres  to  one  son,  263.75  acres  to  another:  how  much  had 
he  left  ? 

23.  What  is  the  difference  between  203.007  and  302.07  ? 
^  24.  Two  men  starting  from  the  same  place,  traveled  in 
<J  opposite  directions,   one  going   571.37    miles,   the   othet 

501.037  miles :  how  much  further  did  one  travel  than  the 
other ;   and  how  far  apart  were  they  ? 

MULTIPLICATION   OF   DECIMALS. 
191.  To  find  the  Product  of  two  or  more   Decimals. 

1.  What  is  the  product  of  45  multiplied  by  .7  ? 
Analysis. — .*]—'h-  Now  -,^,j times  45 =3.15  _ ^i 5 ^ lo,    operation. 

or  31.5,  Ans.     In  the  operation,  we  multiply  by  7  in-  45 

stead  of  1^0  ;  therefore  the  produc*- is  10  times  too  large.  .7 

To  correct  this,  we  point  ofiF  i    figure   on  the   right,  ^j  c  Ans. 
which  divides  it  by  10.    (Arts.  79,  143,  165.) 

2.  What  is  the  product  of  9.7  multiplied  by  .9  ? 
Analysis. — 9.7=^^,  and  .9=A'.    Now  -^^  times  W—    operation. 

fia.=r873-^loo,  or  8.73,  Ans.     In  this  operation   we  9.7 

also  multiply  as  in   whole  numbers,  and   point  off  2  .9 

figures  on  the  right  of  the  product  for  decimals,  which  ~Zj^\  Ans» 
divides  it  by  100.     (Art.  79.) 


150  MULTIPLICATION    OF    DECIMALS. 

Hem. — By  inspecting  these  operations,  we  see  that  e*ch  Aixsr^er 
contains  as  many  decimal  figures  as  there  are  decimal  places  in  both 
factors.     Hence,  the 

KuLE. — Multiply  as  in  loliole  numhers,  and  from  the 
right  of  the  product,  point  off  as  many  figures  for  decimals, 
as  there  are  decimal  places  in  both  factors. 

Notes. — i.  Multiplication  of  Decimals  is  based  upon  the  same 
principles  as  Multiplication  of  Common  Fractions.    (Art.  i66.) 

2.  The  reason  for  pointing  off' the  product  is  this:  The  product  of 
any  two  decimal  numerators  is  as  many  times  too  large  as  there  are 
units  in  the  product  of  their  denominators,  and  pointing  it  off" divides 
it  by  that  product.  (Art.  79.)  For,  the  product  of  the  denominators 
of  two  decimals  is  always  i  with  as  many  ciphers  annexed  as  there 
are  decimal  places  in  both  numerators.     (Arts,  57,  179,  186,  Dem.) 

3.  If  the  produ:/  has  not  as  many  figures  as  there  are  decimals  in 
both  factors,  supply  the  deficiency  by  prefixing  ciphers. 

2.  Multiply  .015  by  .03. 

Solution. — .015  x  .03  =  .00045.  The  product  requires  5  decimal 
places  ;  hence,  3  ciphers  must  be  prefixed  to  45. 


(3-) 

(4.) 

(S-)                (6.) 

Multiply         29.06 

.07213 

.000456                 4360.12 

By                     .005 

.0021 

.0037                         5.000 

7.  4.0005  X  .00301. 

II.  0.0048  X  .0091. 

8.  5.0206  X  4.0007. 

12.     15.004  X. 10009. 

9.  3.0004  X  106. 

13.    6.0103  ^  -00012. 

10.  7.2136  X  100. 

14.  20007  X  .000001, 

15.  If  I  box  contains  17.25  pounds  of  butter,  how  many 
pounds  will  25  boxes  contain  ? 

16.  What  cost  20.5  barrels  of  flour,  at  $10,875  a  barrel? 

17.  If  one  acre  produces  750.5  bushels  of  potatoes,  how 
much  will  .625  acres  produce  ? 

18.  What  cost  53  horses,  at  $200.75  apiece? 

19.  Multiply  28  hundredths  by  45  thousandths. 

20.  What  cost  73.25  yards  of  cJoth,  at  $9,375  per  yard'  ? 

191.  How  are  decimals  multiplied?  How  point  off  the  product  ?  Note.  Ex- 
plain the  reason  for  pointing  off.  If  the  product  does  not  contain  as  many 
figure?  as  there  are  decimals  in  the  factors,  what  is  to  be  done  ?  192.  How  mut 
tiply  \  decimal  by  xo,  loo,  etc. 


MULTIPLICATIOiq"     OF     DECIMALS.  151 

21.  Multiply  5  tenths  by  5  thousandths. 

22.  Multiply  two  hundredths  by  two  ten-thousandths. 

23.  Multiply  seven  hundredths  by  seven  millionlhs. 

24.  Multiply  two  hundred  and  one  thousandths  by  three 
millionths. 

25.  Multiply  five  hundred-thousandths  by  six  thous- 
andths. 

26.  Multiply  four  millionths  by  sixty-three  thousandths. 

27.  Multiply  a  hundred  by  a  hundred-thousandth. 

28.  Multiply  one  million  by  one  millionth. 

29.  Multiply  one  millionth  by  one  billionth. 

192.  To  multiply  Dechnnls  by  10,  100,  1000,  etc. 

30.  Multiply  .43215  by  1000. 

Analysis. — Removing  a  figure  one  place  to  the  left,  we  have  seen 
multiplies  it  by  10.  But  moving  the  decimal  point  one  place  to  the 
right  has  the  same  effect  on  the  position  of  the  figures  ;  therefore,  it 
multiplies  them  by  10.  For  the  same  reason,  moving  the  decimal 
point  two  places  to  the  right  multiplies  the  figures  by  100,  and  so  on. 
In  the  given  example  the  multiplier  is  1000  ;  we  therefore  move  the 
decimal  point  three  places  to  the  right,  and  have  432.15,  the  answer 
required.    Hence,  the 

EuLE. — Move  the  decimal  point  as  many  places  toward 
the  right  as  there  are  ciphers  in  the  multiplier.  (Art.  181.) 

31.  Multiply  32.0505  by  100. 

32.  Multiply  8.00356  by  1000. 

33.  Multiply  0.000243  by  loooo. 

34.  Multiply  0.000058  by  1 00000. 

35.  Multiply  0.000005  by  loooooo. 

^6.  If  a  new^sboy  makes  $0,005  on  eacli  paper,  what  is 
b^'s  profit  on  loooo  papers? 

37.  What  is  the  profit  on  looooo  eggs, at  $0,006  apiece? 

38.  If  a  farmer  gives  4.25  bushels  of  apples- for  one  yard 
of  cloth,  how  many  bushels  should  he  give  for  6.5  yards  ? 

39.  If  a  man  walks  3.75  miles  an  hour,  how  far  will  he 
walk  in  17.5  hours? 


152  DIVISION    OF    DECIMALS. 

DIVISION   OF   DECIMALS. 
193.   To  divide  one  Decimal  by  another. 

1.  How  many  times  .2  in  .8  ? 

Analysis, — These    decimals    have    a    com.  de-         .2). 8 
nom. ;  hence,  we  divide  as  in  Common  Fractions,      A_^o~2  times, 
and  the  quotient  is  a  whole  number.    (Art.  169,  n.) 

2.  How  many  times  .08  in  .7  ? 

Analysis, — Reduced  to  a  common  denominator,  the      .08). 7  000 
given  decimals  become  .08  and  .70.     Now  .08  is  in  .70,      j        o 
8  times,  and  .06  rem.     Put  the  8  in  units'  place.     Re-  *    wo 

duced  to  the  next  lower  order,  .06 =.060,  and  .08  is  in  .060,  .7  of  a 
time,  and,  004  rem.  Put  the  7  in  tenths'  place.  Finally,  ,08  is  in  .0040, 
.05  of  a  time.    Write  the  5  in  hundredths'  place.     Ans.  8.75  times. 

3.  How  many  times  is  .5  contained  in  .025  ? 
Analysis, — Reduced  to  a  common  denominator,     .5oo).025oo 

.5=. 500,  and  .o25=.o25.     Since .500 is  not  contained      ^    „  ooct 
in  ,025,  put  a  cipher  in  units'  place,  and  reduce  to  '     •  j  ' 

the  next  lower  order.  But  ,500  is  not  contained  in  .0250.  Put  a 
cipher  in  tenths'  place,  and  reducing  to  the  next  order,  .500  is  in 
.02500,  ,05  of  a  time.     Write  the  5  in  hundredths'  place. 

Rem, — When  two  decimals  have  a  com.  denom.  the  quotient  fiprures 
thence  arising  are  whole  numbers,  as  in  common  fractions.  (Art.  169,  n.) 

If  ciphers  are  annexed  to  the  remainder,  the  next  quotient  figure 
will  be  tenths,  the  second  hundredths,  &c.     Hence,  the 

KuLE. — Reduce  the  decimals  to  a  cominon  denominator, 
and  divide  the  numerator  of  the  dividend  ly  that  of  the 
divisor,  placing  a  decimal  point  on  the  right  of  the  quotient. 

Annex  ciphers  to  the  remainder,  and  divide  as  before. 
The  figures  on  the  left  of  the  decimal  point  are  whole  num- 
bers ;  those  on  the  right,  decimals. 

Or,  divide  as  in  whole  numbers,  and  from  the  right  of  the 
quotient,  point  off  as  many  figures  for  decimals  as  the  deci' 
mal  places  in  the  dividend  exceed  those  in  the  divisor. 

Notes. — i.  If  there  are  not  figures  enough  in  the  quotient  for  th» 
decimals  required  by  the  second  method,  prejlx  ciphers. 


193,  How  divide  decimals  ?  If  there  is  a  remainder  ?  Hem.  When  two  decimals 
have  a  com,  denom.,  what  is  the  quotient  ?  When  ciphers  are  annexed  to  the 
remainder,  what  ?    When  the  ad  method  is  used,  how  point  oflf  ? 


DlVISIOi^    OF    DECIMALS.  153 

2.  If  there  is  a  remainder  after  the  required  number  of  decimals  is 
found,  annex  the  sign  +  to  the  quotient. 

2.  Divide  .063  by  9.  4.  Divide  642  by  1.07. 

3.  Divide  .856  by  .214.         5.  Divide  4.57  by  11. 

Perform  the  following  dirisions : 

6.  78.4-^-2.6.  14.  .03753^.00006, 

7.  8.45^-3.5-  15-  12^1.2. 

8.  1.262-^-9.7.  16.    I.2-i-.I2. 

9.  .4625-^.65.  17.    .I2-M2. 

10.  97.68-MOO.  18.  .00001-^5. 

11.  6.75-:- 1000.  19.  .00005-^.1. 

12.  .576^10000.  20.  .0003-^.000006. 

13.  45.30-^-3020.  21.  .27-MOOOOOO. 

22.  If  2.25  yards  of  cloth  make  i  coat,  how  many  coata 
can  be  made  of  103.5  yards  ? 

2^.  How  many  rods  in  732.75  feet,  at  16.5  feet  to  a  rod? 

24.  At  $18.75  apiece,  how  many  stoves  can  be  bought 
for  $506.25  ? 

194.  To  Divide  Decimals  by  10,  100,  1000,  etc. 

25.  "What  is  the  quotient  of  846.25  divided  by  100? 

Analysis. — Moving  the  decimal  point  one  place  to  the  left  divides 
a  number  by  10.  For  the  same  reason,  moving  the  decimal  point 
two  places  to  the  left,  divides  it  by  100.  In  the  ppven  question,  re- 
lieving the  decimal  point  two  places  to  the  left,  we  have  8.4625,  the 
answer  required.     Hence,  the 

EuLE. — Move  the  decimal  point  as  many  places  toward 
the  left  as  there  are  ciphers  in  the  divisor.  (Art.  181.) 

26.  Divide  4375.3  by  1000.  28.  Divide  2.53  by  looooo, 

27.  Divid-i  638.45  by  loooo.        29.  Divide  .5 'by  loooooo. 

30.  Bought  1000  pins  for  $.5 :  what  was  the  cost  of  each  ? 

31.  If  a  man  pays  $475  for  loooo  yards  of  muslin,  what 
is  that  a  yard  ? 

194.  How  divide  decimals  by  lo,  loo,  icxxa,  etc.  ? 


UNITED  STATES  MONEY. 

195.  United  States  llouey  is  the  national  cur- 
rency  of  the  United  States,  and  is  often  called  Federal 
Money,  It  is  founded  upon  the  Decimal  Notation,  and  is 
thence  called  Decimal  Currency. 

Its  denominations  are  eagles,  dollars,  dimes,  cents,  and 
mills. 

TABLE. 

10  mills  (m.)        are  i  cent, et 

lo  cents  '*    I  dime, d. 

lo  dimes  "    i  dollar,     -    -  dol.  or  $. 

lo  dollars  "   i  eagle,      .    .    .     .    £J 


NOTATION   OF   UNITED    STATES   MONEY.- 

196.  The  Dollar  is  the  unit;  hence,  dollars  are 
whole  numbers,  and  have  the  sign  (I)  prefixed  to  them. 

In  1 1  there  are  loo  cents;  therefore  cents  are  hun- 
dredths of  a  dollar,  and  occupy  hundredths''  place. 

Again,  in  |i  there  are  looo  mills;  hence,  mills  are 
thousandths  of  a  dollar,  and  occupy  thousandths'  place. 

Notes.— I.  The  origin  of  the  sign  ($)  has  been  variously  ex- 
plained. Some  suppose  it  an  imitation  of  the  two  pillars  of  Her- 
cules, connected  by  a  scroll  found  on  the  old  Spanish  coins.  Others 
think  it  is  a  modified  figure  8,  stamped  upon  these  coins,  denoting 
8  reals,  or  a  dollar. 

A  more  plausible  explanation  is  that  it  is  a  monogram  of  United 
States,  the  curve  of  the  U  being  dropped,  and  the  S  written  over  it. 

195.  What  is  United  States  money?  Upon  what  founded?  Wliat  eometimea 
called?  The  denominations  ?  Repeat  the  Table.  196.  What  is  the  unit?  What  are 
dimes  ?  Cents  ?  Mills  ?  Note.  What  is  the  origin  of  the  sign  $  ?  The  meaning 
of  dime?  Cent?  Mill?  197.  How  write  United  States  money  ?  iVbi^.  How  are 
eagles  and  dimes  expresecd  ?  If  the  number  of  cents  is  less  than  10,  what  must 
be  done  ?    Why  ?    If  the  milb  are  5  or  more,  what  considered  ?    If  less,  what  ? 


UXITED     STATES     MOi^EY.  155 

2.  The  term  Dime  is  the  French  dixieme,  a  tenth;  Cent  from  the 
latin  centum, Q.  hundred;  and  MiU  from  the  Latin  miUe,  a  thousand. 

3.  CJnited  States  money  was  established  by  act  of  Congress,  iu 
1786.     Previous  to  that,  pounds,  shillings,  pence,  etc.,  were  in  use. 

197.  To  express  United  States  money,  decimally. 

Fx.  I.  Let  it  be  required  to  write  75  dollars,  37  cents, 

and  5  mills,  decimally. 

Analysis. — Dollars  are  integers ;  we  therefore  write  the  75  dol- 
lars as  a  whole  number,  prefixing  the  ($),  as  $75.  Again,  cents  are 
hundredths  of  a  dollar  ;  therefore  we  write  the  37  cents  in  the  first 
two  places  on  the  right  of  the  dollars,  with  a  decimal  point  on  theijr 
left  as  $75.37.  Finally,  mills  are  thousandths  of  a  dollar  ;  and  writing 
the  5  mills  in  the  first  place  on  the  right  of  cents,  we  have  $75,375, 
the  decimal  required.     (Art.  179.)     Hence,  the 

EuLE. —  Write  dollars  as  wlwle  numbers,  cents  as  7mn- 
dredths,  and  mills  as  thousandths,  placing  the  sign  (8)  he- 
fore  dollars,  and  a  decimal  point  between  dollars  and  cents. 

Notes. — i.  Eagles  and  dimes  are  not  used  in  business  calcula- 
tions;  the  former  are  expressed  by  tens  of  dollars  ;  the  latter  by  tens 
of  cents.     Thus,  15  eagles  are  $150,  and  6  dimes  are  60  cents. 

2.  As  cents  occupy  two  places,  if  the  number  to  be  expressed  is  less 
than  10,  a  cipher  must  be  prefixed  to  the  figure  denoting  them. 

3.  Cents  are  often  expressed  by  a  common  fraction  having  100  for 
its  denominator.  Thus,  $7.38  is  written  $7A-o>  and  is  read  "  7  and 
',V,j  dollars." 

Mills  also  are  sometimes  expressed  by  a  common  fraction.  Thus, 
12  cts.  and  5  mills  are  written  $0,125,  or  $0.12^ ;  18  cis.  and  7.2  mills 
are  written  $0.1875,  or  $0.18]-,  etc. 

4.  In  business  calculations,  if  the  mills  in  the  result  are  5  or  mor«, 
they  are  considered  a  cent ;  if  less  than  5,  they  are  omitted. 

1.  Write  forty  dollars  and  forty  cents. 

2.  Write  five  dollars,  five  cents  and  five  mills. 

3.  Write  fifty  dollars,  sixty  cents,  and  three  mills. 

4.  Write  one  hundred  dollars,  seven  cents,  five  mills. 

5.  Write  two  thousand  and  one  dollars,  eight  and  a  half 
cents. 

6.  Write  7  hundred  and  5  dollars  and  one  cent. 

7.  Write  84  dollars  and  12}  cents. 

8.  Write  5  and  a  half  cents:  6  and  a  fourth  cents;  11 
aiid  three-ft)urth8  cents,  decimally. 


156  UNITED     STATES     MOKEY. 

9.  Write  7  dollars,  31  and  a  fourth  cents^  decimally. 

10.  Write  19  dollars,  31:1  cents,  decimally. 

11.  Write  14  eagles  and  8  dimes,  decimally. 

12.  Write  5  eagles,  5  dollars,  5  cents,  and  5  mills. 

198.  To  read  United  States  money,  expressed  decimally. 
Rule. —  Gall  the  figures  on  the  left  cf  the  decimal  point, doU 

lars  ;  those  in  the  first  two  places  on  the  right,  cents  ;  the  next 
figure,  mills  ;  the  others,  decimals  of  a  mill. 

The  expression  $37.52748  is  read  37  dollars,  52  cts.,  7 
mills,  and  48  hundredths  of  a  mill. 

Note. — Gents  and  mills  are  sometimes  read  as  decimals  oi  a  do?lar. 
Thus,  $7,225  may  be  read  7  and  225  thousandths  dollars. 

Read  the  following : 

1.  $204.30  5.     $78,104  9.  $1100.001 

2.  $360.05  6.     $90,007  10.  $7.3615 

3.  $500.19  7.  $1001.10  II.  $8.0043 

4.  $61,035  8.  $1010.01  12.  $10.00175 

REDUCTION   OF   UNITED    STATES   MONEY. 
CASE    I. 

199.  To  reduce  Dollars  to  Cents  and  Mills. 

1.  In  $67  how  many  cents? 

Analysis. — As  there  are  100  cents  in  every  opebation. 

dollar,  there  must  be  100  times  as  many  cents        67  X  100  =  6700 
as  dollars  in  the  given  sum.     But  to  multiply        Ans.  6700  cts. 
by  100  wo  annex  two  ciphers.     (Art.  57.) 

2.  In  $84,  how  many  mills? 

Analysis.— For  a  like  reason,  there  are  84  X  1000  =  84000 
1000  times  as  many  mills  as  dollars,  or  10  Ans.  84000  millji. 

times  as  many  mills  as  cents.     Hence,  the 

Rule.— :7b  reduce  dollars  to  cents,  multiply  them  ly  100. 

To  reduce  dollars  to  mills,  multiply  them  hy  1000. 

To  reduce  cents  to  mills,  multiply  them  hy  10. 


198.  How  read  United  States  money  ? 


UN^ITED     STATES     MOl^EY.  157 

Note. — Dollars  and  cents  are  reduced  to  cents ;  also  dollars,  cents, 
and  mills,  to  mills,  by  erasing  the  sign  of  dollars  ($),  and  the  deci- 
mal  point. 

2.  Reduce  $135  to  cents.  6.  Reduce  97  cents  to  mills. 

3.  Reduce  $368  to  mills.  7.  Reduce  $356.25  to  cents. 

4.  Reduce  $100  to  mills.  8.  Reduce  $780,375  to  mills. 

5.  Reduce  $1680  to  cents.  9.  Reduce  I800.60  to  mills. 

CASE     II. 

200.  To  reduce  Cents  and  Mills  to  Dollars. 

I.  In  6837  cents  how  many  dollars? 

Analysis. — Since  100  cents  make  i  dollar,       .  opeeation. 
6837  cents  will  make  as  many  dollars  as  100  is  l)68|37 

contained  times  in   6837,  and    6837^100=68,  $68.37  Ans. 

and  37  cents  over.    (Art.  79.) 

In  like  manner  any  number  of  mills  will  make  as  many  dollars  aa 
1000  is  contained  times  in  that  number.     Hence,  the 

Rule. — To  reduce  cents  to  dollars,  divide  them  ly  100. 

To  reduce  mills  to  dollars,  divide  them  ly  1000. 

To  reduce  mills  to  cents,  divide  them  ly  10.     (Art.  194.) 

Note. — The  first  two  figures  cut  off  on  the  right  are  cents,  the 
next  one  mills. 

2.  Reduce  1625  cts.  to  dols.  6.  Change  89567  cts.  to  dels. 

3.  Reduce  8126  m's  to  dols.  7.  Change  94283  m's  to  dols. 

4.  Reduce  loooo  m's  to  dols.  8.  Change  85600  m's  to  cents. 

5.  Reduce  9265  m's  to  cents.  9.  Change  263475  m's  to  dols. 

10.  A  farmer  sold  763  apples,  at  a  cent  apiece :  how 
many  dollars  did  they  come  to  ? 

n.  A  market  woman  sold  5  hundred  eggs,  at  2  cents 
each:  how  many  dollars  did  she  receive  for  them  ? 

12.  A  fruit  dealer  sold  675  watermelons  at  1000  mills 
apiece :  how  many  dollars  did  he  receive  for  them  ? 

199.  How  reduce  dollars  to  cents  ?  To  mills  ?  How  cents  to  mills  ?  Notg.  How 
dollars  to  cents  and  mills?  300.  How  reduce  cents  and  mills  to  dollars?  NoU. 
VVhot  are  the  figures  cut  off? 


158  U  1^11  ED     STATES     MOKEY. 


ADDITION   OF   UNITED    STATES   MONEY. 

201.  United  States  Money,  we  have  seen,  is  founded 
upon  the  decimal  notation;  hence,  all  its  operations  are 
precisely  the  same  as  the  corresponding  operations  in 
Decimal  Fractions, 

202.  To  find  the  Amount  of  two  or  more  Sums  of  Money, 

1.  Whatis  the  sum  of  $45,625  ;  $109.07;  and  $450,137  ? 

Analysis.  —  Units  of    the  same  order  only  can  be  operation. 

added  together.     For  convenience  in  adding,  we  there-  $45,625 

fore  write  dollars  under  dollars,  cents  under  cents,  etc.,  109,07 

with  the  decimal^  points  in  a  perpendicular  line.     Begin-  45  o.  1 3  7 

ning  at  the   right,   we  add   the  columns  separately,  $604.8-^ 
placing   the  decimal  point  in  the  amount   under  the 

points  in  the  mmibers  added,  to  distinguish  the  dollars  from  cents 
and  mills.     (Art.  197.)     Hence,  the 

KuLE. —  Write  dollars  under  dollars,  cents  under  cents, 
etc.,  and  proceed  as  in  Addition  of  Decimals. 

Note. — If  any  of  the  given  numbers  have  no  cents,  their  place 
ehould  be  supplied  by  ciphers. 

2.  A  man  paid  $13. 62^  for  a  barrel  of  flour,  $25.25  for 
butter,  $9.75  for  coal:  what  did  he  pay  for  all  ? 

3.  A  farmer  sold  a  span  of  horses  for  $457.50,  a  yoke  of 
oxen  for  $235,  and  a  cow  for  $87.75  :  how  much  did  he 
receive  for  all  ? 

4.  What  is  the  sum  of  $97.87-1-;  $82.09;  $2o.i2|? 

5.  What  is  the  sum  of  $81.06;  $69.18;  $67.16;  $7.13? 

6.  What  is  the  sum  of  $101.101 ;  $2io.io|;  $450.27^? 

7.  Add. $7  and  3  cents;  $10;  6\  cents;  i8|  cents. 

8.  Add  $68  and  5  mills ;  87 J  cents;  31 J  cents. 

9.  A  man  paid  $8520.75  for  his  farm,  $1860.45  for  his 
stock,  $1650.45  for  his  house,  and  $1100.07  for  his  furni- 
ture :  what  was  the  cost  of  the  whole  ? 

201.  What  is  said  of  operations  in  United  States  Money?  202.  How  ndd 
TTnited  States  Money?    Note.  If  any  of  the  numbere  hav*  no  centD,  how  proceeds 


Uiq'ITED     STATES     MOis^ET.  1^\) 

10.  A  lady  paid  $31 J  for  a  dress,  $15!-  for  trimmings, 
and  $7-J  for  making:  what  was  the  cost  of  her  dress  ? 

11.  A  grocer  sold  goods  to  one  customer  amounting  to 
$17.50,  to  another  $3o.i8f,  to  another  $2i.o6:J,  and  to 
another  $5 1.73  :  what  amount  did  he  sell  to  all? 

12.  A  young  man  paid  $31.58  for  a  coat;  $11.63  f'^^  ^ 
vest,  $14.11  for  pants,  $10.50  for  boots,  $7 J  for  a  hat,  and 
$1 J  for  gloves :  what  did  his  suit  cost  him  ? 

SUBTRACTION   OF   UNITED    STATES    MONEY. 

203.  To  find  the  jyifference  between  two  Sums  of  Money. 

1.  A  man  having  $1343.87!,  gave  $750.69  to  the  Patriot 
Orphan  Home :  hoAV  much  had  he  left  ? 

Analysis. — Since  the  same  orders  only  can  be  sub-  opiration, 

tracted  one  from  the  other,  for  convenience  we  write  $1343.875 

dollars  under  dollars,  cents  under  cents,  etc.     Begin-  750.69 

ning  at  the  right,  we  subtract  each  figure  separately,  $CQ'?.i8^ 
and  place  the  decimal  point  in  the  remainder  under 
that  in  the  subtrahend,  for  the  same  reason  as  in  subtracting  deci- 
mals.    (Art.  190.)     Hence,  the 

Rule. —  Write  the  less  number  under  the  greater,  dollars 

under  dollars,  cents  under  cents,  etc.,  and  'proceed  as  in 

Subtraction  of  Decimals.     (Art.  190.) 

Note. — If  only  one  of  the  given  numbers  has  cents,  their  place  in 
the  other  should  be  supplied  by  ciphers. 

2.  A  man  having  $861.73,  lost  $328,625  in  gambling: 
how  much  had  he  left  ? 

3.  If  a  man's  income  is  $1750,  and  his  expenses  $1145.3  7  J, 
how  much  does  he  lay  up  ? 

4o  A  gentleman  paid  $1500  for  his  horses,  and  $975  J  for 
his  carriage :  what  was  the  difference  in  the  cost  ? 

5.  A  merchant  paid  $25 73 J  for  a  quantity  of  tea,  and 
sold  it  for  $3158^ :  how  much  did  he  make  ? 

203.  How  8u1)tract  United  States  money?  JVoie.  If  either  number  has  no 
cents,  what  is  to  be  done  ? 


160  U]S^ITED     STATES     MONEY. 

6.  If  I  pay  $5268  for  a  farm,  and  sell  it  for  $4319.67, 
how  much  shall  I  lose  by  the  operation  ? 

7.  From  673  dols.  6 J  cents,  take  501  dols.  and  10  cents. 

8.  From  ion  dols.  12^  cents,  take  600  dols.  and  5  cents. 

9.  From  I  dollar  and  i  cent,  subtract  5  cents  5  mills. 

10.  From  7J  dols.  subtract  7J  cents. 

11.  From  500  dols.  subtract  5  dols.  5  cents  and  5  mills. 

12.  A  3^oung  lady  bought  a  shawl  for  $35  J,  a  dress  for 
^231,  a  hat  for  $iof,  a  pair  of  gloves  for  |ij,  and  gave 
the  clerk  a  hundred  dollar  bill :  how  much  change  ought 
she  to  receive  ? 

MULTIPLICATION  OF  UNITED   STATES  MONEY. 

204.  To  multiply  United  States  Money. 

1.  What  will  9 J  yards  of  velvet  cost,  at  $18 J  per  yard  ? 

Analysis. — 9.^  yards  will  cost  9^  times  as  mucli  as  $18.2  c; 

I    yard.     Now   g\   yds.=9.5   yds.,  and   $i84-=$i8.25.  „_- 

We  multiply  in  tlie  usual  way,  and  since  there  are  

tJiree  decimal  figures  in  both  factors,  we  point  off  three  "     ^ 

in  the  product  for  the  same  reason  as  in  multiplying  ^   5  ^ 

decimals.     (Art.  191.)     Hence,  the  ^173-375 

Rule. — Multiply,  and  jmint  off  the  product,  as  in  Multi- 
plication  of  Decimals.     (Art.  191.) 

Notes. — i.  In  United  States  Money,  as  in  gimple  numbers,  the 
multiplier  must  be  considered  an  abfitract  number. 

2.  If  either  of  the  given  factors  contains  a  common,  fraction,  it  is 
generally  more  convenient  to  change  it  to  a  demnal 

2.  What  will  65  barrels  of  flour  cost,  at  $11.50  a  barrel? 

3.  What  will  145.3  pounds  of  wool  cost,  at  $1.08  a  pound? 

4.  What  cost  75  pair  of  skates,  at  $3.87!-  a  pair  ? 

5.  What  cost  6;^  gallons  of  petroleum,  at  95^  cents 
a  gallon  ? 

6.  At  $4.17^  a  barrel,  what  will  no  barrels  of  apples  cost? 

204.  How  cniltiply  United  States  money  ?    Kote.  What  must  the  multiplier  be  r 


UKITED     STATES     MON^EY.  161 

Perform  the  following  multiplications : 

7.  $511x1.9!-.  II.  $765,401x6.05. 

8.  $6.07-!- X  2.3  J.  12.  $.07  X. 008. 

9.  $10.05  X  6^.  13.  $.005  X  1000. 
10.  $100,031x3.105.                  14.  $1,011  X. 001. 

15.  What  cost  35  pounds  of  raisins,  at  iSJ  cents  a 
pound  ? 

16.  What  cost  51  pounds  of  tea,  at  $1.15 J  a  pound? 

17.  What  cost  150  gallons  of  milk,  at  37 J  cts.  a  gallon  ? 

18.  What  cost  12  dozen  penknives,  at  31 J  cents  apiece? 

19.  What  cost  13  boxes  of  butter,  each  containing  16  J 
pounds,  at  37^  cents  a  pound? 

20.  A  merchant  sold  12  pieces  of  cloth,  each  containing 
35  yards,  at  $4!  a  yard :  what  did  it  come  to  ? 

21.  What  is  the  value  of  21  bags  of  coffee,  each  weigh- 
ing 55  pounds,  at  47^  cents  a  pound? 

22.  What  cost  55  boxes  of  lemons,  at  $3  J  a  box  ? 

2;^.  The  proprietor  of  a  livery  stable  took  37  horses  to 
board,  at  $32  a  month:  how  much  did  he  receive  in  12 
months  ? 

DIVISION    OF   UNITED    STATES    MONEY. 

205.  Division  of  United  States  money,  like  simple  divi- 
sion, embraces  two  classes  of  problems : 

First.  Those  in  wliicli  both  the  divisor  and  dividend  are  money. 

Second.  Those  in  which  the  dividend  is  money  and  the  divisor  is 
an  abstract  number,  or  regarded  as  such. 

In  the  former,  a  given  sum  is  to  be  divided  into  parts,  the  value  of 
each  part  being  equal  to  the  divisor;  and  the  object  is  to  ascertain 
the  number  of  parts.  Hence,  the  quotient  is  times  or  an  abstract 
number,     (Art.  63,  a.) 

In  the  latter,  a  given  sum  is  to  be  divided  into  a  given  number  of 
equal  parts  indic&ted  by  the  divisor ;  and  the  object  \s  to  ascertain 
the  value  or  number  of  dollars  in  each  part.  Hence,  the  quotient  is 
money,  the  same  as  the  dividend.    (Art.  63,  6.) 

ao5.  What  does  DiviBion  of  U.  S.  money  embrace  ?  The  first  class  ?  The  second  f 


162  UNITED     STATES     MOls'EY. 

206.  To  Divide  Money  by  Money,  or  by  an  Abstract  Number 

1.  How  many  barrels  of  flour,  at  $9.56  a  barrel,  can  be 
bought  for  $262.90  ? 

Analysis.— At  $9.56  a  barrel,  $262.90  will  operation. 

6uy  as  many  barrels  as  $956  are  contained  $9.56)$262.9o(27.5 

times  in  $262.90,  or  27.5  barrels.     As  the  191 2 

divisor  and  dividend    both    contain  cents,  T^^io 

tliey  are  the  same  denomination  ;  therefore,  6692 

the  quotient  27,  is  a  whole  number.     Annex-  3— 

.1  1  .1        1  47 oO 

mg  a  cipher  to  the  remamder,  the  next  quo-  '  o 

tient  figure    is    tenths.     (Art.    193,    Rem)  -zZ 

Hence,  the 

Rule. — Divide,  and  point  off  the  quotient,  as  in  Division 
of  Decimals.     (Arts.  64,  193.) 

Notes. — i.  In  business  matters  it  is  rarely  necessary  to  carry  the 
quotient  beyond  mills. 

2.  If  there  is  a  remainder  after  all  the  figures  of  the  dividend  have 
been  divided,  annex  ciphers  and  continue  the  division  as  far  as  de- 
sirable, considering  the  ciphers  annexed  as  decimals  of  the  dividend. 

2.  If  67^  ca,ps  cost  1 1 23.75,  what  will  i  cap  cost? 

3.  If  165  lemons  cost  $8.25,  w^hatwill  i  lemon  cost? 
^4.  Paid  I852.50  for  310  sheep  :  what  Tfas  that  apiece? 
L  5.  If  356  bridles  cost  $1040,  what  will  i  bridle  cost? 

Find  the  results  of  the  following  divisions : 

6.  $17.50-:- 1. 1 75  10.  $.oo5^$.o5 

7.  $365.07 -^  $1.01  II.   $ioi-^$i.oi 

8.  $1000-7-25  cts.  12.  $5oo-^$.o5  ^^ 

9.  $.25H-$25  13.    $l200-^$.002  ^ 

14.  If  410  chairs  cost  $1216,  what  will  i  chair  cost? 

15.  At  $9j  a  ton,  how  much  coal  will  $8560  buy? 

16.  When  potash  is  $120.35  per  ton,  how  much  can  be 
bought  for  $35267.28? 

17.  If  2516  oranges  cost  $157.25,  what  will  one  cost? 
t8.  Paid  $273.58  for  5000  acres  of  land:  what  was  that 

per  acre  ? 

2o6.  How  divide  money  ?    NoU.  If  there  is  a  remainder,  how  proceed  ? 


UNITED     STATES     MOifEY. 


163 


COUNTING-ROOM    EXERCISES. 

207.  The  Ledger  is  the  principal  look  of  account? 
kept  by  business  men.  It  contains  a  Irief  record  of  their 
monetary  transactions.  All  the  items  of  the  Bay  Booh  are 
transferred  to  it  in  a  condensed  form,  for  reference  and 
preservation,  the  debits  (marked  Dr.)  being  placed  on  the 
left,  and  the  credits  (marked  Or.)  on  the  right  side. 

208.  Balaneing  an  Aecotcnt  is  finding  the  differ- 
ence between  the  debits  and  credits. 

Balance  the  following  Ledger  Accounts : 


(!•) 


(2.) 


Dr. 

Cr. 

Dr. 

Or. 

$8645.23 

$2347.19 

$76421.26 

$43261.47 

160.03 

141.07 

2406.71 

728.23 

2731.40 

2137.21 

724.05 

6243.41 

4242.25 

3401.70 

86025.21 

75.69 

324-31 

2217.49 

9307.60 

53268.75 

3313-17 

168.03 

685.17 

3102.84 

429.18 

329.17 

61.21 

456.61 

4536.20 

2334-67 

7824.28 

32921.70 

641.46 

4506.41 

60708.19 

6242.09 

182.78 

239.06 

1764.85 

20374-34 

2634.29 

2067.12 

32846.39 

290.25 

3727-34 

675-89 

385-72 

4536.68 

840.68 

1431.07 

23.64 

42937-74 

6219.77 

4804.31 

6072.77 

819.35 

4727.91 

6536.48 

50641.39 

30769.27 

23-45 

720.34 

2062.40 

8506.35 

650.80 

36.45 

301-53 

97030.48 

3267.03 

7050.63 

26.47 

876.20 

680.47 

904.38 

805.03 

89030.50 

64.38 

78.05 

2403.67 

384.06 

8350.60 

307-63 

10708.79 

70207.48 

207,  What  is  the  ledger  ?  What  does  it  contain  ?   IIow  balance  an  account  ? 


164 


BILLS, 


MAKING    OUT    BILLS. 

209.  A  Bill  is  a  tvriUen  statement  of  goods  sold,  ser- 
vices rendered,  etc.,  and  should  always  include  the  price 
of  each  item,  the  date,  and  the  place  of  the  transaction. 

A  Bill  is  Beceiptedf  when  the  person  to  whom  it 
is  due,  or  his  agent,  writes  on  it  the  words  "Received 
payment,"  and  his  name. 

209,  a,  A  Statement  of  Account  is  a  copy  of  the 
items  of  its  cleMts  and  credits. 

Notes. — i.  The  abbreviation  Dr.  denotes  debit  or  debtor ;  Or., 
credit  or  creditor ;  per,  by  ;  the  character  @,  stands  for  at.  Thus,  5 
books,  @  3  shillings,  signifies,  5  books  the  price  of  which  is  3  shil- 
lings apiece. 

2.  The  learner  should  carefully  observe  the  form  of  Bills,  tho 
l^lace  where  the  date  and  the  names  of  the  buyer  and  seller  are 
placed,  the  arrangement  of  the  items,  etc. 

Copy  and  find  the  amount  due  on  the  following  Bills : 


i\\  Thtladsi.fbia,  March  zst,  18 j2. 

Hon.  Heney  Barkard, 

To  J.  B.  Lippii^coTT  &  Co.,  Dr. 


For  6  Webster's  Dictionaries,  4to.,  @  $12.50 
"    8  reams  paper,                            @    $3-75 
"  36  slates,                                        @    $0.27 

$75 

30 

9 

00 
00 

72 

$114 

$84 
$30 

72 

Credit. 

By  10  School  Architecture,               (S    $5.45 
"     8  Journals  of  Education,           @    $3.75 

Balance, 

$54 

50 

GO 

50 

209,  What  is  a  bill  ?  209,  a.  A  statement  of  account  ?  Where  is  the  date 
placed?  (See  form.)  The  name  of  buyer?  Seller?  How  is  the  payment  of  a 
bill  shown  ?    JVote.  What  does  Dr.  denote  ?    Or.  ? 


(2.) 


BILLS.  165 

New  Yobk,  April  2d,  1871. 


Mrs.  W.  C.  Gabfield, 


Bought  of  A.  T.  Stewaet. 


28  yds.  silk, 

35  yds.  table  linen, 

6  pair  gloves, 
43?  yds.  muslin, 

I  do3.  pr.  cotton  hose. 

Amount. 


@  $3-50 
@,  $2.12^ 

@$i.75 
@  $0.33 
@,  $0.80 


Eeceived  Payment, 


A.  T.  Stewart. 


\3v  New  Orleans,  May  5th,  1872. 

George  Peabody,  Esq., 

Bought  of  Jacob  Barker. 


Feb. 

I 

.^7 

Marc 

I13 

25 

April 

L30 

75  bis.  pork, 
160  bis.  flour. 
500  gals,  molasses, 

75  boxes  raisins, 
256  gals,  kerosene. 


@  $25.00 

@    $8.75 
%    $0.93 

@    $5,371- 
@    $0.87^ 


Amount, 


Received  Payment, 


J.  Barker, 
By  John  Howard. 


fA\  Boston,  May  23<f,  1871. 

Messrs.  Fairfield  &  Webster, 

To  H.  W.  Hall,  Dr. 


t'or    319  yds.  broadcloth, 
'•     416  yds.  cassimere, 
"    mo  yds.  muslin, 
**      265  yds.  ticking, 


$5-87i 
$2.10 
$0.28 
$0.47 


Amount, 


Ueceir^  Payment  by  Note, 

George  Anderson, 

For  H.  W.  Hali. 


166  BILLS. 

5.  James  Brewster  bought  of  Horace  Foote  &  Co.,  New- 
Haven,  May  16th,,  1867,  the  following  items:  175  pounds 
of  sugar,  at  1 7  cents ;  5  gallons  of  molasses,  at  6^  cents ; 
3  boxes  of  raisins,  at  $6i;  15  pounds  of  tea,  at  |ij:  what- 
was  the  amount  of  his  bill? 

6.  George  Bliss  &  Co.  bought  of  James  Henry,  Cincin- 
nati, June  3d,  1867,  1625  bushels  of  wheat,  at  $1.95;  130 
barrels  of  flour,  at  $11;  265  pounds  of  tobacco,  at  48 
cents;  and  1730  pounds  of  cotton,  at  27^  cents:  what  was 
the  amount  of  their  bill  ? 

6jj  7.  Pinkney  &  Brother  sold  to  Henry  Kutledge,  Rich- 
m.ond,  July  15th,  1867,  i  shawl,  $450;  19  yards  of  silk,  at 
$S'^S ;  16  yards  point  lace,  at  81 1 ;  6  pair  gloves,  at  $2.05  ; 
and  12  pair  hose,  at  87I  cents:  required  the  amount 

8.  Bought  37  Greek  Readers,  at  1 1.85;  60  Greek 
Grammars,  at  $1.45;  75  Latin  Grammars,  at  I1.38;  25 
Virgils,  at  I3.62  ;  14  Hiad,  at  $3.28 :  required  the  amount. 

BUSINESS   METHODS. 

210.  Calculations  in  United  States  money  are  the  same 
as  those  in  Decimals  ;  consequently,  in  ordinary  business 
transactions  no  additional  rules  are  required.  By  Reflec- 
tion and  Analysis,  however,  the  operations  may  often  be 
greatly  abbreviated.     (Arts.  100,  21 1-2 14.) 

1.  What  will  265  hats  cost,  at  I7  apiece  ? 

Analysis. — 265  hats  will  coat  265  times  as  much  as  one  hat ;  and 
$7x265  =  11855. 

Or  thus :  At  $1  each,  265  hats  will  cost  $265  ;  hence,  at  $7,  they 
will  cost  7  times  $265,  and  $265  x  7 =$185  5. 

Notes, — i .  Here  the  price  of  one  and  the  number  of  articles  are  given, 
to  find  their  cost;  hence,  it  is  a  question  in  Multiplication.  (Art.  51,) 

2.  A  mechanic  sold  12  ploughs  for  $114:  what  was  the 
price  of  each  ? 

Analysis. — The  price  of  i  plough  is  ^^  as  much  as  that  of  13 
ploughs;  and  $II4-Hi2=$9.50. 


COUKTIKG-ROOM     EXERCISES.  167 

Note, — 2.  Here  the  number  of  articles  and  their  cost  are  g^ven, 
to  find  the  price  of  one,  which  is  simply  a  question  in  Division. 

3.  A  farmer  paid  $921.25  for  a  number  of  sheep  valued 
at  82.75  apiece:  how  many  did  he  buy? 

Analysts. — At  $2.75  apiece,  $921.25  will  buv  as  many  sheep  as 
$2.75  is  contained  times  in  $921.25  ;  and  $921.25-^12.75  =  335. 

Note. — 3.  Here  the  whole  cost  and  the  price  of  one  are  given,  to 
find  the  number,  which  is  also  a  question  in  Division. 

4.  What  will  287  chairs  cost,  at  I2 J  apiece  ? 

5.  What  will  75  sofas  cost,  at  I57.50  apiece? 

6.  AYhat  will  350  table  books  cost,  at  12J  cents  apiece? 

7.  What  cost  119  tons  of  coal,  at  $7^  a  ton  ? 

8.  Paid  I84  for  252  pounds  of  butter:  what  was  that  a 
pound  ? 

9.  If  SJ  cords  of  wood  cost  $46.06,  what  will  i  cord  cost? 

10.  If  46  acres  of  land  are  worth  $1449,  what  is  i  acre 
worth  ? 

11.  How  many  Bibles  at  $5 J  can  be  bought  for  $561  ? 

12.  How  many  vests  at  $8^-,  can  be  bought  for  $1262.25  ? 

13.  How  much  will  it  cost  a  man  a  year  for  cigars, 
allowing  he  smokes  5  a  day,  averagmg  6^  cenljs  each,  and 
365  days  to  a  year  ? 

14.  A  lady  bought  18  yards  of  silk,  at  $2.12  J  a  3^ard;  14 
yards  of  delaine  at  6;^  cents;  12  skeins  of  silk,  at  6^  cents 
a  skein :  what  was  the  amount  of  her  bill  ? 

15.  What  will  it  cost  to  build  a  railroad  265  miles  long, 
at  $11350  per  mile? 

16.  A  farmer  sold  his  butter  at  34  cents  a  pound,  and 
received  for  it  $321.67^:  how  many  pounds  did  he  sell? 

17.  The  cheese  made  of  the  milk  of  53  cows  in  a  season 
was  sold  for  $1579.40,  at  20  cents  a  pound:  how  many 
pounds  were  sold ;  and  what  the  average  per  cow  ? 

18.  A  merchant  sold  3  pieces  of  muslin,  each  containing 
45  yards,  at  40^  cents  a  yard,  and  took  his  pay  in  wheat 
at  $i\  a  bushel:  how  much  wheat  did  he  receive? 


168  COUKTINO-ROOM     EXERCISES. 

19.  If  a  man  earns  $9^  a  week,  and  spends  $4 J,  ho^a 
much  will  he  lay  up  in  5  2  weeks  ? 

20.  If  a  man  drinks  3  glasses  of  liquor  a  day,  costing  10 
cents  a  glass,  and  smokes  5  cigars  at  6  cents  each,  how 
tiiany  acres  of  land,  at  $ij  an  acre,  could  he  buy  for  the 
Bum  he  pays  for  liquor  and  cigars  in  35  years,  allowing 
365  days  to  a  year  ? 

21.  A  grocer  bought  175  boxes  of  oranges,  at  $6.37}, 
and  sold  the  lot  for  $637.50 :  what  did  he  make,  or  lose  ? 

211.  To  find  the  Cost  of  a  number  of  articles,  the  Price  of 
one  being  an  Aliquot  part  of  $1. 

I.  What  will  168  melons  cost,  at  i2|-  cents  apiece  ? 


OPMUTION. 


Analysis.— By  inspection  the  learner  will  perceive 
that  12^  cents =^  dollar.     Now,  at  i  dollar  apiece, 
168  melons  will  cost  $168.     Bat  the  price  is  i  of  a      8)168 
dollar  apiece;   therefore,  they  will  cost  \  of  $168,  %2i  Ans. 

which  is  $21.     Hence,  the 

Rule. — Take  such  a  part  of  the  given  number  as  indi- 
cated by  the  aliquot  part  of  ^1,  expressing  the  price  of  one; 
the  result  will  be  the  cost.     (Arts.  105,  270.) 

2.  "What  will  265  pounds  of  raisins  cost,  at  25  cents  ? 

3.  "What  cost  195  pounds  of  butter,  at  ssi  cents  ? 

4.  "What  cost  352  skeins  of  silk,  at  6 J  cents? 

5.  What  cost  819  shad,  at  50  cents  apiece? 

6.  At  i2|-  cents  a  dozen,  what  will  100  dozen  eggs  cost  ? 

7.  At  20  cents  a  yard,  what  will  750  yards  of  calico  cost  ? 

8.  At  i6|  cents,  what  will  1250  pine  apples  come  to? 

9.  What  cost  1745  yards  of  delaine,  at  33 J  cents  ? 

10.  At  8 J  cents  apiece,  what  will  375  pencils  cost? 

11.  At  25  cents  apiece,  what  will  11 67  slates  cost? 

1 2.  How  much  will  175  dozen  eggs  cost,  at  20  cts.  a  doz.  ? 

13.  What  will  219  slates  cost,  at  12^  cents  each  ? 

14.  At  i6f  cts.  apiece,  what  will  645  melons  cost? 

15.  At  33  J  cts.  apiece,  how  much  will  347  penknives  cost  ? 


COUKTIKa-ROOM     EXERCISES.  169 

212.  To  find  the  Cost  of  a  number  of  articles,  the  Price  of 

one  being  $1  plus  an  Aliquot  part  of  $1. 

i6.  What  cost  275  pounds  of  tea,  at  $1.25  a  pound? 

Analysis, — The  price  of  i  pound  $1.25=$!  +  $^.    At       4)275 
$1  a  pound,  the  cost  of  275  pounds  would  be  $275.    But  68^7 

the  price  is  $1  +  $.ir ;   therefore  275    pounds  will  cost     ^ —^ 

once  $275  +  i  of  $275  =  $343.75-     Hence,  the  «>343-75 

EuLE. — To  the  number  of  articles,  add  \,  |,  J  of  itself, 
as  the  case  may  he;  the  sum  will  he  their  cost. 

17.  At  $1.25  per  acre,  what  will  168  acres  of  land  cost? 

18.  At  $1.50  apiece,  what  will  365  chairs  cost  ? 

19.  What  cost  512  caps,  at  %i^  apiece  ? 

20.  At  $i.i6|  apiece,  what  cost  12  dozen  fans? 

21.  A  man  sold  200  overcoats  at  a  profit  of  $1.20  apiece : 
how  much  did  he  make  ? 

213.  To  find  the  Number  of  articles,  the  Cost  being  given, 

and  the  Price  of  one  an  Aliquot  part  of  $1. 

22.  How  many  spellers,  at  12^  cents  apiece,  can  he 
bought  for  $25  ? 

Analysis. — 12^  cents=$|;  therefore  $25  will     $.i2j=$|^ 
buy  as  many  spellers  as  there  are  eighths  in  $25,     $25-^$-|-=20o 
and  25-f-i=200.    Ans.  200  spellers.    Hence,  the 

Rule. — Divide  the  cost  of  the  whole  hy  the  aliquot  part 
of%i,  expressing  the  price  of  one. 

23.  How  man}^  yards  of  flannel,  at  50  cents,  can  be 
bought  for  $6.83  ? 

24.  How  many  pounds  of  candy,  at  33^  centig,  can  be 
purchased  with  $375  ? 

25.  Paid  $450  for  cocoa-nuts  which  were  25  cents  each: 
how  many  Avere  bought  ? 

26.  At  20  vients  each,  how  many  pine-apples  can  be  pur- 
chased for  $538  ? 

211.  How  find  the  cos.""  of  articles  by  aliquot  parts  of  $i  ?  212.  How  when  the 
price  is  $1  plus  an  aliquot  part  of  $1?  213.  How  find  the  number  of  articleb, 
when  cost  is  given,  and  the  price  of  one  is  an  aliquot  part  of  $1  ? 

8 


170  COUNTIIf  G-EOOM     EXERCISES- 

214.  To  find  the  Cost  of  articles,  sold  by  the  100,  or  1000. 

1.  What  will  1765  oranges  cost,  at  $6,125  a  hundred? 

Analysis. — At    $6,125    for    each    orauge,    1765         operation. 
oranges  would  cost  1765  times  $6  125,  and  $6,125  x  $6,125 

1765  =  $10810.625.     But   the  price  is  $6,125  for  a  1765 

hundred ;   therefore  this  product  is  100  times  too       $108.10621; 
large.     To  correct  it,  we  divide  by  100,  or  remove 
the  decimal  point  two  places  to  the  left.    (Art.  181.) 

In  like  manner  if  the  price  is  given  by  the  1000,  we  multiply  the 
'price  and  number  of  articles  together,  and  remove  the  decimal  point 
in  the  product  three  places  to  the  left,  which  divides  it  by  1000. 
Hence,  the 

Rule.  —  Multij^tly  the  price  and  niimher  of  articles 
together,  and  divide  the  product  ly  100  or  1000,  as  the 
case  may  require,     {xi  rt,  181 .) 

Note. — In  business  transactions,  the  letter  C  is  sometimes  puv 
for  hundred;  and  M  for  thousand. 

2.  What  will  4532  bricks  cost,  at  I17.25  per  M.  ? 

3.  What  cost  1925  pounds  of  maple  sugar,  at  I12.50  pev 
hundred? 

4.  What  cost  25268  feet  of  boards,  at  $31.25  per  thous- 
and? 

5.  At  $5f  per  hundred,  how  much  will  20345  pounds 
of  flour  come  to  ? 

6.  At  I6.25  per  hundred,  what  will  19263  pounds  of 
codilsh  come  to  ? 

7.  What  cost  10250  envelopes,  at  I3.95  per  thousand? 

8.  What  cost  1275  oysters,  at  $1.75  per  hundred? 

9.  What  cost  13456  shingles,  at  $7.45  per  M.  ? 

10.  What  cost  82  rails,  at  $5  J  a  hundred  ? 

11.  What  cost  93  pine  apples,  at  $15.25  a  hundred? 

12.  What  cost  355  feet  of  lumber,  at  $45  per  thousand^ 


214.  How  find  the  cost  of  articles  sold  by  the  100  or  1000  ?    Note.  What  does  C 
stand  for?     WhatM? 


COMPOUND    NUMBERS. 

215.  Slmjyle  J^umbers  are  those  which  contain 
units  of  one  denomination  only;  as,  three,  live,  2  oranges, 
4  feet,  etc.     (Art.  loi.) 

216.  Compound  JV^umbers  are  those  which  con- 
tain units  of  two  or  more  denominations  of  the  sat7ie 
nature;  as,  5  pounds  and  8  ounces;  3  yards,  2  feet  and  4 
inches,  etc. 

But  the  expression  2  feet  and  4  pounds,  is  not  a  com- 
pound number;  for,  its  units  are  of  unliJce  nature. 

Notes. — i.  Compound  Numhers  are  restricted  to  the  divisions  of 
Money,  Weights,  and  Measures,  and  are  often  called  Denominate 
Nunibers. 

2.  For  convenience  of  reference,  tlie  Compound  Tables  are  placed 
togrether.  If  the  teacher  wishes  to  give  exercises  upon  them  as  they 
are  recited,  he  will  find  examples  arranged  in  groups  <x>i responding 
with  the  order  of  the  Tables  in  Arts,   276,  279. 


MOIfEY. 

217.  MJonej/  is  the  measure  or  standard  of  value.  It 
is  often  called  currency,  or  circulating  medium,  and  is  of 
two  kinds,  metallic  and  paper. 

Metallic  Money  consists  o^  stamped  pieces  of  metal, 
called  coins.    It  is  also  called  specie,  or  specie  currency. 

I^aper  Money  consists  of  notes  or  hills  issued  by  the 
Government  and  Banks,  redeemable  in  coin.  It  is  often 
called  paper  currency. 

U15.  What  are  simple  numbere  ?  216.  Compound?  Give  an  example  of  each. 
Nd^e.  To  what  s.-e  compound  numbers  restricted?  217,  What  is  money?  Me- 
tallic money  ?    Paper  money  ? 


172  COMPOUND     NUMBERS. 


UNITED  STATES  MONEY. 

218.  United  States  Money  is  the  national  cur- 
realty  of  the  United  States,  and  is  often  called  Federal 
Money.  Its  denominations  are  eagles,  dollars,  dimes,  cents, 
and  mills.. 

TABLE. 

lo  mills  (m)  are  i  cent, ct. 

lo  cents  "    I  dime,  .        _        .        -        -     e?. 

lo  dimes         "    i  dollar,  _        _         -         -  dol.  or  $. 

lo  dollars       "    i  eagle E. 

219.  The  Metallic  Currency  of  the  U7iited  States 
consists  0^ gold  and  silver  coins,  and  the  minor  coins.* 

1.  The  gold  coins  are  tlie  double  eagle,  eagle,  half  eagle,  quarter 
eagle,  three  dollar  piece,  and  dollar. 

The  dollar,  at  the  standard  weight,  is  the  unit  oi  value. 

2.  The  silver  coins  are  the  dollar,  "trade"  dollar,  hMf  dollar, 
quarter  dollar,  and  dime. 

3.  The  minor  coins  are  the  s-<^ent  and  2>-cent  pieces,  and  the  cent. 

220.  The  weight  and  purity  of  the  coins  of  the 
United  States  are  regulated  by  the  laws  of  Congi-ess. 

1.  The  standard  weight  of  the  gold  dollar  is  25.8  gr. ;  of  the  quar- 
ter eagle,  64  5  gr. ;  of  the  3-dollar  piece,  77.4  gr. ;  of  the  half  eagle, 
129  gr. ;  of  the  eagle,  258  gr. ;  of  the  double  eagle,  516  gr. 

2.  'W\Qice'ght  of  the  dollar  is  412^  grains,  Troy;  the  "trade" 
dollar,  420  grains  ;  the  half  dollar,  \i\  grams  ;  the  quarter  dollar 
and  dime,  one-half  and  one-fifth  the  weight  of  the  half  dollar. 

3.  The  weight  of  the  5-ccnt  piece  is  77. 16  grains,  or  5  grams ;  of 
the  3-cent  piece,  30  grains ;  of  the  cent,  48  grains. 

4.  The  standard  purity  of  the  gold  and  silver  coins  is  nine-tenths 
pure  metal,  and  one-tenth  alloy.  The  alloy  oi  gold  coins  is  silver  AxiA. 
copper  ;  the  silver,  by  law,  is  not  to  exceed  one-tenth  of  the  whole 
alloy.     The  alloy  of  silver  coins  is  pure  copper. 

219.  Of  what  does  the  metallic  currency  of  the  United  States  consist?  What 
are  the  gold  coins  ?  The  silver  ?  The  minor  coins  ?  220.  How  is  the  weight  and 
purity  of  United  States  coins  regulated  ? 

*  Act  of  Congrass,  March  3d,  1873. 


COMPOUl!^D     NUMBERS.  173 

5.  The  five-cent  and  three-cent  pieces  are  composed  of  one-fourth 
nickel  and  three-fourths  copper ;  the  cent,  of  95  parts  copper  and  5 
parts  of  tin  and  zinc.     They  are  known  as  nickel  and  bronze  coins. 

Notes. — i.  The  Trade  dollar  is  so  called,  because  of  its  intended 
use  for  commercial  purposes  among  the  great  Eastern  nations. 

2.  The  gold  coins  are  a  legal  tender  in  all  payments;  the  siher  coins, 
for  any  amount  not  exceeding  $5  in  any  one  payment ;  the  minor 
coins,  for  any  amount  not  exceeding  25  cents  in  any  one  payment. 

3.  The  diameter  of  the  nickel  5-cent  piece  is  two  centimeters,  and 
its  weight  5  grams.  These  magnitudes  present  a  simple  relation  of 
the  Metric  weights  and  measures  to  our  own. 

4.  The  silver  5-cent  and  3-cent  pieces,  the  bronze  2-cent  piece, 
the  old  copper  cent  and  half -cent  are  no  longer  issued.  Mills  were 
never  coined. 

221.  The  I*aper  Currency  of  the  United  States 
consists  of  Treasury-notes  issued  by  the  Government 
known  as  Greenbacks,  and  Banh-notes  issued  by  Banks. 

Note. — Treasury  notes  less  than  $1,  are  called  Fractional  Cur- 
rency. ■ 

ENGLISH   MONEY. 

222.  English  3Ioney  is  the  national  currency  of 
Great  Britain,  and  is  often  called  Sterling  Money.  The 
denominations  sltq  poimds,  shillinys,  pence,  and  fa?  thinys. 

TABLE. 

4  farthings  (gr.  or /ar.)  are  I  penny, -    -  d. 

12  pence  "    i  shilling,    - s. 

20  shillings  "    I  pound  or  sovereign,    -     -  £ 

21  shillings  "    i  guinea, g. 

Notes. — i.  The  gold  coins  are  the  sovereign  and  lialf  sovereign. 
The  pound  sterling  was  never  coined.  It  is  a  bank  note,  and  is 
represented  oj  the  sovereign.  Its  legal  value  as  fixed  by  Congress  is 
$4.8665.     This  is  its  intrinsic  value,  as  estimated  at  the  U.  S.  Mint. 

What  is  the  alloy  of  gold  coins  ?  Of  silver  ?  221.  Of  what  does  U.  S.  paper 
carrency  cousist?  222.  English  money?  The  denominations?  The  Table? 
Jfote.  Is  the  pound  a  coin  ?    How  represented  ?    What  is  the  value  of  a  pound  ? 


174  COMPOUIS^D    NUMBERS. 

2.  The  silver  coins  are  the  crown  (5s.) ;  the  half-crown  (28.  6d.) . 
the  florin  (2s.) ;  the  shilling  (i2d.) ;  the  six-penny,  four-penny,  and 
three-penny  pieces. 

3.  The  copper  coins  are  the  penny,  half-penny,  and  farthing. 

4.  Farthings  are  commonly  expressed  asfractions  of  a  penny,  as  y^d. 

5.  The  oblique  mark  (/)  sometimes  placed  between  shillings  and 
pence,  is  a  modification  of  the  long/. 

CANADA    MONEY. 

223.  Canada  3Io7iey  is  the  legal  currency  of  the 
Dominion  of  Canada.  Its  denominations  are  dollars, 
cents,  and  mills,  which  have  the  same  value  as  the  cor- 
responding denominations  of  IT.  S.  money.  Hence,  all  tlie 
oj)erations  in  it  are  the  same  as  those  in  U.  S.  money. 

Note. — The  present  system  was  established  in  1858. 

FRENCH     MONEY. 

224.  French  Money  is  the  national  currency  of 
France.  The  denominations  are  the  franc,  the  decime, 
and  centime, 

TABLE. 

10  centimes        .        _        .        _    are  i  decTme. 
10  decimes  -        -        -        -       "    i  franc. 

Notes. — i.  The  system  is  founded  upon  the  decimal  notation  ; 
hence,  all  the  operations  in  it  are  the  same  as  those  in  U.  S.  money, 

2.  The  franc  is  the  unit ;  decimes  are  tentJis  of  a  franc,  and 
centimes  hundredths. 

3.  Centimes  by  contraction  are  commonly  called  cents. 

4.  Decimes,  like  our  dimes,  are  not  used  in  business  calculations; 
they  are  expressed  by  tens  of  centimes.  Thus,  5  decimes  are  ex- 
jressed  by  50  centimes;  63  francs,  5  decimes,  and  4  centimes  are 
written,  63.54  francs. 

5.  The  legal  value  of  i\\Q  franc  in  estimating  duties,  is  19.3  cents, 
its  intrinsic  value  being  the  same. 

Shilling  ?  Florin  ?  Crown  ?  How  are  farthings  often  written  ?  223.  What  is 
Canada  money?  Its  denominations  ?  Their  value?  224.  French  money  ?  lis 
denomiuationa  ?    The  Table  ?    mte.  The  unit  ?    The  value  of  a  franc  ? 


WEIGHTS. 

225.  WeigJit  is  a  measure  of  the  force  called  gravity^ 
by  which  all  bodies  iend  toward  the  center  of  the  earth. 

226.  Net  Weight  is  the  weight  of  goods  without  the 
bag,  cask,  or  box  which  contains  them. 

Gross  Weight  is  the  weight  of  goods  with  the 
bag,  cask,  or  box   in   which  they  are   contained. 

The  weights  in  use  are  of  three  kinds,  yiz  :  Troy,  Avoir- 
dupois, and  Apothecaries^  Weight. 

TROY    WEIGHT. 

227.  Troy  Weight  is  employed  in  weighing  gold, 
silver,  und  jewels.  The  denominations  ixyq  pounds,  ounces* 
pennyioeights,  and  grains. 

TABLE. 

24  grains    {gr)     are  i  pennyweiglit,    -         -    -    pwt, 

20  pennyweights    "    i  ounce, oz. 

12  ounces  "    i  pound, lb. 

Note. — The  unit  commonly  employed  in  weighing  diamonds, 
pearls,  and  otheT  jewels,  is  the  carat,  which  is  equal  to  4  grains. 

228.  The  Standard  Unit  of  weight  in  the  IJnited 
States,  is  the  Troy  pound,  which  is  equal  to  22.794377  cubic 
inches  of  distilled  water,  at  its  maximum  density  (39.83° 
Fahrenheit),*  the  barometer  standing  at  30  inches.  It  is 
exactly  equal  to  the  Imperial  Troy  pound  of  England,  the 
former  being  copied  from  the  latter  by  Captain  Kater.f 

Note. — The  original  element  of  weight  is  a  grain  of  wheat  taken 
from  the  middle  of  the  ear  or  head.  Hence  the  name  grain  as  a 
unit  of  weight. 

225.  What  is  weight  ?  226.  Troy  weight  ?  The  denominations  ?  The  table  ? 
227.  The  standard  nnit  of  weight  ?  Note.  The  original  element  of  weight  ? 
2»8.  Avoirdupois  weight ?    The  denominations ?    The  Table? 

*  Hassler.  t  Professor  A.  D.  Bache. 


176  COMPOUND    KUMBEES. 


AVOIRDUPOIS   WEIGHT. 

229.  Avoirdupois  Weight  is  used  in  weighing 
all  coarse  articles ;  as,  hay,  cotton,  meat,  groceries,  etc., 
and  all  metals,  except  ^o?<i  and  silver.  The  denominations 
are  tons,  hundreds,  pounds ^  and  ounces, 

TABLE. 

i6  ounces  (02.)     -    -   are  i  pound,      .-..-.  ^. 

100  pounds,  -    -    -    .     "     I  hundredweight,     -    -    -  twt. 

20  cwt.,  or  2000  lbs.,     "     I  ton, T. 

The  following  denominations  are  sometimes  used : 

1000  ounces  are  i  cubic  foot  of  water. 

100  pounds   "  I  quintal  of  dry  fish. 

196  pounds   "  I  barrel  of  floar, 

200  pounds  "  I  barrel  of  fish,  beef,  or  pork. 

280  poimds   "  I  barrel  of  salt. 

Notes. — i.  The  ounce  is  often  divided  into  halves,  quarters,  etc. 

2.  In  business  transactions,  the  dram,  the  quarter  of  25  lbs.,  and 
i\\Q  firkin  of  56  lbs.,  are  not  used  as  units  of  Avoirdupois  Weight. 

Rem. — ^In  calculating  duties,  the  law  allows  112  pounds  to  a 
hundredweight,  and  custom  allows  the  same  in  weighing  a  few 
coarse  articles ;  as,  coal  at  the  mines,  chalk  in  ballast,  etc.  In  all 
departments  of  trade,  however,  both  custom  and  the  law  of  most  of 
the  States,  call  100  pounds  a  Jmndredweight 

230.  The  Standard  Avoirdupois  pound  is  equal 
to  7000  grains  Troy,  or  the  weight  of  27.7015  cubic  inches 
of  distilled  water,  at  its  maximum  density  (39.83°  Fah.),  the 
barometer  being  at  30  inches.  It  is  equal  to  the  Imperial 
Avoirdupois  pound  of  England. 

231.  Comparison  of  Avoirdupois  and  Troy  Weight 

7000  grains  equal  i  lb.  Avoirdupois. 

5760       "  "       I  lb.  Troy. 

437^       "  "       I  oz.  Avoirdupois. 

480         "  "I  oz.  Troy. 


229.  To  what  is  the  Avoirdupois  pound  equal?    230.  What  is  net  weight? 
Gross  weight? 


COMPOUND     NUMBEKS.  177 

APOTHECARIES'     WEIGHT. 

232.  Apothecaries'  Weight  is  used  by  pbysiciana 
in  prescribing,  and  apothecaries  in  mixing,  dry  medicines. 

20  grains  (^r.)  are  I  scruple,    -    -    -    -  sc,  ot  3. 

3  scruples         "    i  dram, dr.,  or  3  • 

8  drams  "    i  ounce,       -    .    -    -  oz.,  or  3 . 

12  ounces  "    i  pound,      -     -    -    -  lb.,  or  ft . 

Note, — The    only  difference    between    Troy  and  Apothecaries' 

weight  is  in  the  subdivision  of  the  ounce.    The  pound,  ounce,  and 
grain  are  the  same  in  each. 

232,  a.  Apothecaries'  Fluid  Pleasure  is  used 
in  mixing  liquid  medicines. 

60  minims,  or  drops  (Hl  or  gtt.)  are  i  fluid  drachm,    -    -  /3  . 
8  fluid  drachms  "    i  fluid  ounce,       -    -  /§  . 

16  fluid  ounces  "    i  pint, 0. 

8  pints  "    I  gallon,  -----    Cong. 

Note. — Gtt.  for  giittae,  Latin,  signifying  drops ;    0,  for  octarius, 
Latin  for  one-eighth ;  and  Cong,  congiarium,  Latin  for  gallon. 


MEASUEES   OF  EXTEl^SIOK 

233.  Extension  is  that  which  has  one  or  more  of 
the  dimensions,  length,  breadth,  or  thickness;  as,  line.s^ 
surfaces,  and  solids. 

A  line  is  that  which  has  length  without  hreadth. 

A  surface  is  that  which  has  length  and  hreadth  with- 
out thickness. 

A  solid  is  tliat  which  has  length,  hreadth,  and  thickness. 

Note. — A  measure  is  a  conventional  standard  or  unit  by  which 
values,  weights,  lines,  surfaces,  solids,  etc.,  are  computed. 

234.  The  Standard  Unit  of  length  is  the  yard,  which  is  de- 
termined from  the  scale  of  Troughton,  at  the  temperature  of  62° 
Fahrenheit.     It  is  equal  to  the  British  Imperial  yard. 


232.  In  what  19  Apothecaries'  Weight  used  ?  Table  ?  233.  What  is  extension  ? 
Aline?  A  surface?  A  solid?  Mote.  A  measure?  224.  What  is  the  standara 
unit  of  length  ? 


178  COMPOUND    NUMBERS. 

LINEAR   MEASURE. 
235.  Linear  lleasiire  is  used  in  measuring  that 
which  has  length  without  breadth;  as,   lines,   distances. 
It  is  often  called  Long  Measure.     The  denominations  are 
leagues,  miles, furlongs,  rods,  yards,  feet,  and  inches. 

TABLE. 

12  inches  {in.)      are  i  foot, ft. 

3  feet  "    I  yard, yd. 

S^  yds.  OT  i6}  ft.   "    I  rod,  perch,  or  pole,    -    -    -  r.  or  p. 
40  rods  "    I  furlong, fur. 

8  fur.  or  320  rods  "    i  mile, .    .  m. 

3  miles  "    I  league, I. 

The  following  denominations  are  used  in  certain  cases : 

4  in.  =  I  hand,  for  measuring  the  height  of  horses. 
9  in.  =  I  span. 

18  in.  =  I  cubit. 

6  ft.  =1  fathom,  for  measuring  depths  at  sea. 
3.3  ft.  =  I  pace,*  for  measuring  approximate  distances. 

5  pa.  =  I  rod,  "  "  " 
i|-statute  mi.  :=  i  geographic  or  nautical  mile. 

60  geographic,  or  )  t  ^.t  x 

69,^Jtatute  m.  nearly,  f  =  ^  ^^^^^^  «^  ^^^  ^^^^t«^- 
360  degrees  =  i  circumference  of  the  earth. 

A  knot,  used  in  measuring  distances  at  sea,  is  equivalent  to  a  nau- 
tical mile. 

(For  the  English  and  French  methods  of  determining  the  standard 
of  length,  see  Higher  Arith.) 

Notes. — i.  The  original  element  of  linear  measure  is  a  grain  or 
kernel  of  barley.  Thus,  3  barley-corns  were  called  an  inch.  But 
the  barley-corn,  as  a  measure  of  length,  has  fallen  into  disuse. 

2.  The  inch  is  commonly  divided  into  halves,  fourths,  eighths,  or 
tenths  ;  sometimes  into  twelfths,  called  lines. 

3.  The  mile  of  the  Table  is  the  common  land  mile,  and  contains 
5280  ft.  It  is  called  the  statute  mile,  because  it  is  recognized  by  law, 
both  in  the  United  States  and  England. 

235.  For  what  is  linear  measure  used  ?    The  denominations  ?    The  Table  f 
Note.  The  original  element  of  linear  measure  ?  How  is  the  inch  commonly  divided  ? 
♦  A  military  pace  or  step  is  variously  estimated  2I  and  3  fe«t. 


COMPOUND     NUMBERS.  179 

236.  The  Linear  Unit  employed  by  surveyors  is 
Gimter's  Cham,  which  is  4  rods  or  66  ft.  long,  and  is  sub- 
iivided  as  follows : 

7.92  inclies  (i?i.)  are  i  link, I. 

25  links  ''    I  rod  or  pole   -     -    -  r. 

4  rods  "    I  cliain, ch. 

80  chains  "    i  mile m. 

Note. — Gunter's  chain  is  so  called  from  the  name  of  its  inventor. 
Engineers  of  the  present  day  commonly  use  a  chain,  or  measuring 
tape  100  feet  long,  each  foot  being  divided  into  tenths. 

CLOTH     MEASURE. 

237.  Cloth  lleasure  is  used  in  measuring  those 
articles  of  commerce  whose  length  07ily  is  considered ;  as, 
cloths,  laces,  ribbons,  etc.  Its  principal  unit  is  the  linear 
yard.  This  is  divided  into  halves,  quarters,  eighths,  and 
sixteenths. 

TABLE. 


3  ft.  or  36  in.  are     i  yard,         .     -     - 

yd. 

18  in.,                  "       I  half  yard      -     - 

-    kyd. 

9  in.,                  "       I  quarter  yard     - 

-    \yd. 

42  in.,                  "       I  eighth 

-    Ujd. 

2^:  in.,                  "       I  sixteenth " 

-  -h  yd. 

Notes. — r.  The  old  Ells  Flemish,  English,  and  French,  are  no 
longer  used  in  the  United  States ;  and  the  nail  {2\  inches),  as  a  unit 
of  measure,  is  practically  obsolete. 

2.  In  calculating  duties  at  the  Custom  Houses,  the  yard  is  divided 
into  tenths  and  hundredths. 

SQUARE    MEASURE. 

238.  Square  Pleasure  is  used  in  measuring  sur- 
faces, or  that  which  has  le7igth  and  Ireadth  without  thick- 
ness ;  as,  land,  flooring,  etc.  Hence,  it  is  often  called 
land  or  surface  measure.  The  denominations  are  acres, 
square  rods,  square  yards,  square  feet,  and  square  inches. 


236.  What  is  the  linear  unit  commonly  employed  by  eurveyor°  ?  2^7.  Cloth 
aieasure  ?  Its  principal  unit  ?  How  i?  the  yard  divided  ?  The  Table  ?  238.  Square 
measure?    The  denominations?    TheTablo? 


180 


COMPOUND     NUMBEKS. 


TABLE. 

144  square  inches  [sq.  in.)  sire  i  square  foot,     -    -  sg.  ft. 
9  square  feet  "    i  square  yard,    -    -  sq.  yd. 

2o\  sq.  yards,  or )  «,    j  i   sq.  rod,   perch 

272^  sq.  feet,         )  I      or  pole,    -    -    -  sq.  r. 

160  square  rods  "    i  acre,  -....-  A. 

640  acres  "    i  square  mile,   -     -  sq.  m, 

239.  The  Unit  of  Land  Measure  is  the  Acre. 
and  is  subdivided  as  follows : 


-  P- 


625  sq.  links  are  i  pole  or  sq.  rod,  ■ 

16  poles  "    I  square  chain.    -  -        -    .sq.  c. 

losq.  chains,  or)    „  . 

i6osq.  rods  [         ^  ^^^^'  -        -  -        -    ^. 

Notes. — i.  The  Rood  of  40  sq.  rods  is  no  longer  used  as  a  unit  of 
measure, 

2.  A  Square,  in  Architecture,  is  100  square  feet. 

239,  a.  The  public  lands  of  the  United  States  are  divided  into 
Tovmships,  Sections,  and  Quarter-sections. 

A  Township  is  6  miles  square,  and  contains  36  sq.  miles. 

A  Section  is  i  mile  square,  and  contains  640  acres. 

A  Quarter-section  is  160  rods  square,  and  contains  160  acres. 

240.  A  Squai^e  is  a  rectilinear  figure 
which  has  four  equal  sides,  and  four  right 
angles.    Thus, 

A  Square  Inch  is  a  square,  each  side 
of  which  is  i  inch  in  length. 

A  Square  Yard  is  a  square,  each  side 
of  which  is  I  yard  in  length. 

This  measure  is  called  Square  Measure, 
because  its  measuring  unit  is  a  square. 


g  eq.  ft.  =  i  sq.  yd. 


241.  A  Ilectangle  or  Rectangular  Figure  is  one  which 
has/owr  sides  and  four  right  angles.  When  all  the  sides  are  equal, 
it  is  called  a  square  ;  when  the  opposite  sides  only  are  equal,  it  is 
called  an  oblong  or  parallelogram. 

242.  The  Area  of  a  figure  is  the  quantity  of  surface  it  contains, 
and  Ig  often  called  its  superficial  contents. 


Note.  How  are  the  Gorenimeiit  lands  divided  ?  How  much  land  in  a  town- 
iibip  ?  In  a  section  ?  In  a  quarter-section  ?  240.  What  is  a  square  ?  A  square 
inch  ?  A  pquare  yard  ?  Why  is  square  measure  so  called  ?  241.  What  is  a  recfr 
auirular  figure  ?    242.  What  is  the  area  of  a  figure  ? 


COMPOUI^^D     i^UMBERS, 


181 


243.  The  area  of  all  rectangular  surfaces  is  found  by  multiplying 
the  length  and  breadth  together. 

244.  The  area  and  one  side  being  given,  the  other  side  is  found 
J  dividing  the  area  by  the  given  side.  (Art.  93.) 


CUBIC     MEASURE. 

245.  Cubic  Measure  is  used  in  measuring  solids, 
or  that  which  has  length,  breadth,  and  thickness;  as, 
timber,  boxes  of  goods,  the  capacity  of  rooms,  ships,  etc. 
Hence  it  is  often  called  Solid  Measure.  The  denomina- 
tions are  cords,  tons,  cubic  yards,  cubic  feet,  and  cubic  inches. 


TABLE. 

1728  cubic  inches  {cii.  in)  are  i  cubic  foot,       cu.  ft. 
27  cubic  feet  "    i  cubic  yard,      cu.  yd. 

128  cubic  feet  "    i  cord  of  wood,  C. 

245,  fi»  A  Cord  of  wood  is  a  pile  8  ft.  long,  4  ft.  wide.,  and  4 
ft.  high  ;  for  8  x  4  x  4=128. 

A  Cord  Foot  is  one  foot  in  length  of  such  a  pile ;  hence,  8  cord 
ft.=i  cord  of  wood. 

245,  b.  A  Kegister  Ton  is  the  standard  for  estimating  the 
capacity  or  tonnage  of  vessels,  and  is  100  cu.  ft. 

A  Shipping  Ton,  used  in  estimating  cargoes,  in  the  U.  S.,  is 
40  cu.  ft. ;  in  England,  42  cu.  ft. 

Note. — The  ton  of  40  ft.  o(  round,  or  50  ft.  of  heion  timber  is  sel. 
dom  or  never  used. 


246.  A  Cube  is  a  regular  solid  bounded 
by  six  equal  squares  called  its  faces.  Hence, 
its  length,  breadth,  and  thickness  are  equal 
to  each  other.     Thus, 

A  Cubic  Inch  is  a  cube,  each  side  of 
which  is  a  square  inch  ;  a  Cubic  Yard  is  a 
cube,  each  side  of  which  is  a'square  yard,  etc. 

This  Measure  is  called  Cubic  Measure, 
because  its  measuring  unit  is  a  cube. 


3x3x3=27  cu.  ft. 


/      .''       /      ^ 

^ 

'i___^___^|| 

0 

ilii 

n,  i,:^-M.:    :     l>r 

243.  How  is  the  area  of  rectangular  surfaces  found  ?  245.  Cubic  measure  ?  The 
denominations?  Table?  ^ote.  Describe  a  cord  of  wood?  A  cord  foot?  A 
register  ton  ?    Shipping  ton  ?    346.  What  is  a  cube  ?    A  cubic  inch  ?    Yard  ? 


182  COMPOUND     IfUMBERS. 

247.  -^  rectangular  body  is  one  bounded  by  six  rectangular  sides, 
eack  opposite  pair  being  equal  and  parallel;  as,  boxes  of  goods, 
blocks  of  hewn  stone,  etc. 

Wtien  all  the  sides  are  equal,  it  is  called  a  cuhe  ;  when  the  opposite 
gides  only  are  equal,  it  is  called  a  parallelopiped. 

248.  The  contents  or  solidity  of  a  body  is  the  quantity 
of  matter  or  5j??ace  it  contains. 

249  The  contents  of  a  rectangular  solid  are  found,  by 
multijjlying  the  length,  hreadth,  and  thichness  together. 


MEASUEES    OF    OAPAOITT. 

250.  The  capacity  of  a  vessel  is  the  quantity  of  space 
included  within  its  limits. 

Pleasures  of  Capacity  are  divided  into  two  classes, 
iry  and  liquid  measures. 

DRY     MEASURE. 

251.  l^ry  JMeasure  is  used  in  measuring  grain, 
fruit,  salt,  etc.  The  denominations  are  chaldrons,  bushels, 
jJGcks,  quarts,  and^JiVz^^. 

TABLE. 

2  pints  {pt.)    -     -    are  I  quart, (fi. 

8  quarts      -     .     -       "    i  peck, ph. 

4  pecks,  or  32  qts.,     "    i  bushel, hu. 

36  bushels   -    -     -       "    I  chaldron,      ....  cli. 

252.  The  Standard  Unit  of  Dry  Measure  is  the 
bushel,  which  contains  2150.4  cubic  inches,  or  77.6274  lbs. 
avoirdupois  of  distilled  water,  at  its  maximum  densit}'. 

247.  What  is  a  rectangular  body  ?  When  all  the  sides  are  equal,  what  called  ? 
When  the  opposite  sides  only  are  equal?  248.  What  are  the  contents  of  a  solid 
holy?  24Q.  How  find  the  contents  of  a  rectangular  solid?  2co.  The  caparitv 
of  a  vessel?  2i?i.  Dry  measure?  The  denominations ?  The  Table?  252.  The 
standard  unit  of  dry  measure? 


COMPOUND     NUMBERS. 


183 


It  is  a  cylinder  i8J  in.  in  diameter,  and  8  in.  deep,  the 
same  as  the  old  Winchester  bushel  of  England.*  The 
British  Imperial  Bushel  contains  2218.192  cu.  inches. 

Notes. — i.  The  dry  quart  is  equal  to  i^  liquid  quart  nearly. 
2.  Tlie  chaldron  is  used  for  measuring  coke  and  bituminous  coal. 

2E3.  The  Standard  Bushel  of  different  kinds  of  grain,  seeds, 
etc.,  acc-o^ding  to  the  laws  of  New  York,  is  equal  to  the  following 
number  of  pounds : 


32  lbs.  =  I  bu.  of  oats. 

58  lbs. 

=  I  bu.  of  corn. 

44  lbs.  =  I      "      Timothy  seed. 

( wheat,     peas, 

„  ,-                 ,M  buckwheat,    or 

60  lbs. 

=  I      "    •<     potatoes,  or 

48  lbs.  =  I      "1      barley. 

(     clover-seed. 

55  lbs.  E=  I      "     flax-seed. 

56  lbs.  =  I      "     rye. 

62  lbs. 
100  lbs. 

_         u   J  beans,  or  blue- 
""              (      grass  seed. 
=  I  cental  of  grain. 

Notes. — i.  The  cental  is  a  standard  recently  recommended  by 
the  Boards  of  Trade  in  New  York,  Cincinnati,  Chicago,  and  other 
large  cities,  for  estimating  grain,  seeds,  etc.  Were  this  standard  gen- 
erally adopted,  the  discrepancies  of  the  present  system  of  grain  deal- 
ing would  be  avoided. 

2.  Bushels  are  changed  to  centals,  by  multiplying  them  by  the 
nufnber  of  pounds  in  one  bushel,  and  dividing  the  product  by  100. 
The  remainder  will  be  hundredths  of  a  cental. 


LIQUID     MEASURE. 

254.  Liquid  Pleasure  is  used  in  measuring  milk, 
wine,  vinegar,  molasses,  etc.,  and  is  often  called  Wine 
Measure.  The  denominations  are  liogslieads,  larrels,  gal- 
lons, quarts,  innts,  and  gills, 

TABLE. 

4  gills  {gi^  are  i  pint, pt. 

2  pints  "  I  quart, qt. 

4  quarts         "  i  gallon.  ...    -     .    -  gal. 

31.^  gallons       "  I  barrel, bar.  or  bbl 

63  gallons        "  I  hogshead,       -    -    -     -  Jihd. 

254.  Liquid  measure  ?    The  denominations  ?    Table  ? 
*  Professor  A.  D.  Bache. 


184  COMPOUJS^D     LUMBERS. 

255.  The  Standard  Unit  of  Liquid  Measure  is  tha 
gallon,  which  contains  231  cubic  inches,  or  8.338  lbs. 
avoirdupois  of  distilled  water,  at  its  maximum  densit}^. 
The  British  Imperial  Gallon  contains  277.274  cu.  inches. 

Notes. — i.  The  barrel  and  hogshead,  as  units  of  measure,  are 
chiefly  used  in  estimating  the  contents  of  cisterns,  reservoirs,  etc 

2.  Beer  Measure  is  practically  obsolete  in  this  country.  The  old 
beer  gallon  contained  282  cubic  inches. 

CIRCULAR    MEASURE. 

256.  Circular  Pleasure  is  used  in  measuring 
angles,  land,  latitude  and  longitude,  the  motion  of  the 
heavenly  bodies,  etc.  It  is  often  called  Angular  Measure. 
The  denominations  are  signs,  degrees,  minutes,  and  seconds. 

TABLE. 

60  seconds  (")  are  i  minute,         -  -  -  ' 

60  minutes         "    i  degree,         -  -  -  o  ,  or  deg. 

30  degrees         "    i  sign,     -        -  -  -  s. 

12  signs,  or  360°  "    i  circumference,  -  dr. 

Note. — Signs  are  used  in  Astronomy  as  a  measure  of  the  Zodiac 

257.  A  Circle  is  a  plane  figure  bounded  by  a  curve  line,  every 
part  of  which  is  equally  distant  from  a  point  within  called  the 
center. 

The  Circumference  of  a  circle  is  the  curve  line  by  which  it  is 
bounded. 

The  Diameter  is  a  straight  line  drawn  through  the  center,  ter- 
minating at  each  end  in  the  circu7nference. 

The  Radius  is  a  straight  line  drawn 
from  the  center  to  the  circumference,  and 
is  equal  to  half  the  diameter. 

An  Arc  is  any  part  of  the  circumfer- 
ence. 

In  the  adjacent  figure,  A  D  E  B  F  is 
the  circumference  ;  C  the  center ;  A  B  the 
diameter ;  C  A,  C  D,  C  E,  etc.,  are  radii ; 
A  D,  D  E,  etc.,  are  arcs. 


255.  The  standard  of  Liquid  meaeure  ?  Note.  What  of  Beer  measure  ?  256.  In 
what  is  Circular  measure  used  ?  The  denominations  ?  The  Table  ?  357.  WM 
111  a  circle  ?    The  circumference  ?    Diameter  ?    Radius  ?    An  arc  f 


COMPOUND     i^UMBEKS.  18^ 

258.  A  Plane  Angle  is  the  quantity  of  divergence  of  two 
straight  lines  starting  from  the  same  point.  ^ 

The  Lines   wliich   form   the   angle  are 
called  the  sides,  and  the  point  from  which 
they  start,  the  vertex.     Thus,  A  is  the  ver- 
tex of  the  angle  B  A  C,  A  B  and  A  C  the  .-^ 
sides.                                                                              A^      ^"- 

259.  A  JPerpendicular  is  a  straight 
line  which  meets  another  straight  line  so  as 
to  make  the  two  adjacent  angles  equal  to  '\ 
each  other,  as  A  B  C,  A  B  D.  J 

Each  of  the  two  lines  thus  meeting  is  | 

perpendicular  to  the  other.  ^ 

CUD 

260-  A  Rif/ht  Angle  is  one  of  the  two  equal  angles  formed 
by  the  meeting  of  two  straight  lines  which  are  perpendicular  to 
each  other.    All  other  angles  are  called  oblique. 

260.  a.  The  Measure  of  an  angle  is  the  arc  of  a  circle  included 
between  its  two  sides,  as  the  arc  D  E,  in  Fig,,  Art.  258. 

261.  A  Degree  is  one  360th  part  of  the  circumference  of  a  circle. 
It  is  divided  into  60  equal  parts,  called  minutes ^  the  minute  is 
divided  into  60  seconds,  etc.  Hence,  the  length  of  a  degree,  minute, 
etc.,  varies  according  to  the  magnitude  of  different  circles. 

The  length  of  a  degree  of  longitude  at  the  equator,  also  the  aver- 
age length  of  a  degree  of  latitude,  adopted  by  the  U.  S,  Coast  Sur- 
vey, is  69.16  statute  miles.     At  the  latitude  of  30°  it  is  59.81  miles, 
at  60°  it  is  34.53  miles,  and  at  90°  it  is  nothing.* 

262.  A  Semi -cir cum fei'ence  is  half  a  circumference,  or  180°. 
A  Quadrant  is  onefourth  of  a  circumference,  or  90°. 

263.  If  two  diameters  are  drawn  perpendicular  to  each  other, 
they  will  form  four  right  angles  at  the  center,  and  divide  the  cir- 
cumference into  four  equal  parts.     Hence, 

A  right  angle  contains  90° ;  for  the  quadrant,  which  measures  it, 
is  an  arc  of  90°. 

258.  A  plane  angle?  The  sides  ?  The  vertex?  259.  A  perpendicular ?  260.  A 
right  angle  ?  260,  a.  What  is  the  measure  of  an  angle  ?  261.  What  is  a  degree? 
Upon  what  does  the  length  of  a  degree  depend  ?  What  its  length  at  the  equator  ? 
262.  What  is  a  semi-circumference  ?  How  many  degrees  does  it  contain  ?  A 
quadrant?  263.  If  two  diameters  are  drawn  perpendicular  to  each  other,  what 
is  the  result  ?    How  many  degrees  in  a  right  angle  ? 

*  Encyclopedia  Britannica. 


18G 


COMPOUND     NUMBERS. 


MEASUREMENT    OF    TIME. 

264.  Time  is  a  portion  of  duration.  It  is  divided 
into  centuries,  years,  months,  iveelcs,  days,  hours,  minutes* 
and  seconds. 

TABLE. 


60  seconds  {sec.)                  are 

I  minute 

-     .    - 

-    m. 

60  minutes                             " 

I  hour    - 

. 

-    h. 

24  hours                                " 

I  day      - 

- 

-    d. 

7  days 

I  week  . 

- 

-    w. 

365  days,  or        )                     ,< 
52  w.  and  I  d.  i 

I  common  year 

-    c.p 

366  days                                   " 

I  leap  year 

- 

-   I.y- 

12  calendar  months  {mo.)    " 

I  civil  year 

- 

-  y- 

100  years 

I  century 

„ 

-    c. 

WOTE. — In  most  business  transactions  30  days  are  considered  a 
monih.     Four  weeks  are  sometimes  called  a  lunar  month. 

265.  A  Civil  Year  is  the  year  adopted  by  govern- 
ment for  the  computation  of  time,  and  includes  both  com- 
mon  and  leap  years  as  they  occur.  It  is  divided  into  12 
calendar  months,  as  follows : 


January  (Jan.)  ist  mo.,  31  d. 

r'ebruary  (Feb.)  2d      "  28  d. 

March  (Mar.)  3d      "  31  d. 

April  (Apr.)  4th    "  30  d. 

May  (May)  5th    "  31  d. 

June  (Jime)  6th    "  30  d. 


July  (July)  7th  mo.,  31  d. 

Aug-ust  (Aug.)  8th  "  31  d. 
September  (Sep.)  9th  "  30  d. 
October  (Oct.)  loth  "  31  d. 
November  (Nov.)  nth  "  30  d. 
December  (Dec.)  12th    "     31  d. 


Notes. — i.  The  following  couplet  will  aid  the  learner  in  remem^ 
bering  the  months  that  have  30  days  each  : 

"  Thirty  days  hath  September, 
April,  June,  aud  November." 

All  the  other  months  have  31  days,  except  February,  which  in 
common  years  has  28  days ;  in  leap  years,  29. 

266.  Time  is  naturally  divided  into  days  and  years. 
The  former  are  measured  by  the  revolution  of  the  earth  on 
its  axis ;  the  latter  by  its  revolution  around  the  sun. 


264.  What  is  time  ?    The  denominations  ?    The  Table  f 


COMPOUifD     NUMBERS.  187 

267.  A  Solar  year  is  the  time  in  which  the  earth, 
starting  from  one  of  the  tropics  or  equinoctial  points,  re- 
volves around  the  sun,  and  returns  to  the  same  point.  It 
is  thence  called  the  tropical  or  equinoctial  year,  and  is 
equal  to  z^S^-  sh.  48m.  49.7  sec* 

Note. — i.  The  excess  of  tlie  solar  above  tlie  common  year  is  6 
hours  ot  \  of  a  day,  nearly ;  hence,  in  4  years  it  amounts  to  about  i 
day.  To  provide  for  this  excess,  i  day  is  added  to  every  4th  year, 
which  is  called  Leap  year  or  Bissextile.  This  additional  day  is  given 
to  February,  because  it  is  the  shortest  month. 

2.  Leap  year  is  caused  by  the  excess  of  a  solar  above  a  comm&ti 
year,  and  is  so  called  because  it  leaps  over  the  limit,  or  runs  on  i  day 
more  than  a  common  year. 

3.  Every  year  that  is  exactly  divisible  by  4,  except  centenrJal 
years,  is  a  Leap  year  ;  the  others  are  common  years.  Thus,  1868,  '72, 
etc.,  are  leap  years ;  1869,  '70,  '71,  are  common.  Every  centennial 
year  exactly  divisible  by  400  is  a  leap  year ;  the  other  centennir.l 
years  are  common.  Thus,  1600  and  2000  are  leap  years;  1700,  1800, 
and  1900  are  common. 

268.  An  Apparent  Solar  Day  is  the  time  hetwccn 
the  apparent  departure  of  the  sun  from  a  given  meridian 
and  his  return  to  it,  and  is  shown  by  sun  dials. 

A  True  or  Mean  solar  day  is  the  average  length  of 
apparent  solar  days,  and  is  divided  into  24  equal  parts, 
called  hours,  as  shown  by  a  perfect  clock. 

269.  A  Civil  Day  is  the  day  adopted  by  govern- 
ments for  business  purposes,  and  corresponds  with  the 
mean  solar  day.  In  most  countries  it  begins  and  ends  at 
midnight,  and  is  divided  into  two  parts  of  12  hours  each; 
the  former  being  designated  A.  m.;  the  latter,  p.  m. 

Note.  A  meridian  is  an  imaginary  circle  on  the  surface  of  the 
earth,  passing  through  the  poles,  perpendicular  to  the  equator. 
A.  M.  is  an  abbreviation  of  ante  meridies,  before  midday ;  p.  m.,  of 
post  meridies,  after  midday. 

266-  How  is  time  natarally  divided  ?    How  is  the  former  caused  ?   The  latter  ? 
*  Laplace,  Somerville,  Baily's  Tables. 


188 


COMPOUND  l^UMBEES. 


MISCELLANEOUS  TABLES 


12  tilings  are  i  dozen. 
12  doz.        "    I  gross. 

£4  sheets  are  i  quire  of  paper. 
20  quires  "    i  ream. 

2  leaves  are  i  folio. 

4  leaves   "    i  quarto  or  4to. 

8  leaves   "    i  octavo  or  8vo. 


12  gross      are  i  great  gross. 
20  things    "    I  score. 

2  reams      are  i  bundle. 
5  bundles    "    i  bale. 

12  leaves  are  i  duodecimo  or  i2ma 
1 8  leaves  "    i  eighteen  mo. 
24  leaves   "    i  twenty-four  mo. 


Note. — The  terms  folio,  quarto,  octavo,  etc.,  denote  the  number 
©f  leaves  into  which  a  sheet  of  paper  is  folded  in  making  books. 


270.  Aliquot  Parts  of  a  Dollar,  or  100  cents. 

12^  cents  =  $^ 
10    cents  =  $iV 
8^  cents  =  $iV 


50  cents  =  %\ 
33;^  cents  =  $;^ 
25  cents  =  $1" 
20  cents  ■=  %s 
i6|  cents  =  $^ 


6^  cents  =  $1^^ 
5    cents  ==  liV 


271.  Aliquot  Parts  of  a  Pound  Sterling, 


loshil.  =£,\ 

6s.  8d.  =--  £:V 

5  shil.  =  £L 

4shil.  =£^ 


3S.  4d. 

2S.  6d. 


IS.  8d.  =  £  ,V 
272.  Aliquot  Parts  of  a  Pound  Avoirdupois. 


12  ounces  =  f  pound. 
8       "        =i       " 
5^      "        -^      " 


4  ounces  \  pound. 
2      "       i      " 
I       "        h    " 


273.  Aliquot  Parts  of  a  Year. 


9  months  =  \  year. 
8      "        =  f    " 
6      "        =i     " 
4      "        =^    " 


3  months 

2       "         =  t 

I       "         =  ,J 


i  year. 


274.  Aliquot  Parts  of  a  Month. 

5  days  =    \  mentis. 

3  "  =-h  " 
2  "  =,V  " 
I    "     =^cr      " 


20  days  =  §  month. 

15    "     =i      " 

10    "     =  i      " 

6     "      =  ^      " 


REDUCTION. 

275=  Hediiction  is  changing  a  number  from  one 
denomination  to  another,  without  altering  its  value.  It  is 
either  descending  or  ascending. 

Reduction  Descending  is  changing  higher  de- 
nominations to  loiver ;  as,  yards  to  feet,  etc. 

Reduction  Ascending  is  changing  lower  denomi- 
nations to  higher;  as,  feet  to  yards,  etc. 

276.  To  reduce  Higher  Denominations  to  Lower, 

1.  How  many  farthings  are  there  in  £23,  7s.  sJd.  ? 
Analysis.— Since  there  arc  20s.  in  a  operation. 

pound,  there  must  be  20  times  as  many  £23,  78.  5d..  I  far. 

shillings  as  pounds, p^i^s  the  given  shil-  20 

lings.     Now   20  times   23   are  460,   and  "~T~ 

460s. +  78. =4673.     Again,  since  there  are  t 

I2d.   in    a   shilling,    there    must   be    12        

times  as  many  pence  as  shillings, p?w«  the  S^'^9  d. 

given  pence.     But  12  times  467  are  5604,     4 

and   '56o4d.  +  5d,  =  56o9d.     Finally,   since        22437  f^^*  ^ns. 
there  are  4  farthings  in  a  penny,  there 

must  be  4  times  as  many  farthings  as  pence,  plus  the  given  farthings. 
Now  4  times  5609  are  22436,  and  22436  far. +  1  far.  =  22437  far 
Therefore,  in  £23,  7s.  s^d.  there  are  22437  far.     Hence,  the 

EuLE. — Multiply  the  highest  denomination  hy  the  numher 
required  of  the  next  lower  to  make  a  unit  of  the  higher,  and 
to  the  product  add  the  lower  denorni nation. 

Proceed  in  this  manner  tvith  the  successive  denomina- 
tions, till  the  one  required  is  reached. 

2.  How  many  pence  in  £8,  los.  7d.  ?         Ans.  2047d. 

3.  How  many  farthings  in  12s.  gd.  2  far.? 

4.  How  many  farthings  in  £41,  5s.  4icL.? 

275.  What  is  Reduction?  How  many  kinds?  Descending?  Ascendinc? 
276.  How  are  higlier  denominations  reduced  to  lower?  Explain  Ex.  i  from  th« 
blackboard  ? 


190  REDUCTION. 

277.  To  reduce  Lower  Denominations  to  Higher, 

5.  In  22437  farthings,  how  many  pounds,  BhiUin^s, 
pence,  and  farthings  ? 

Analysis. — Since  in  4  farthings  there  operation. 

is  I  penny,  in  22437  farthings  there  are  4)22437  far. 

as  many  pence  as  4  farthings  are  con-  i2V^6oQd.  I  far. 

tained    times    in    22437    farthings,    or  "wT         ^ 

5609  pence  and  i  farthing  over.     Again,  }^   '      5    • 

since  in  12  pence  there  is  i  shilling,  in  -£23,  7§. 

5609  pence  thert^  are  as  many  shillings  A71S.  £23,  7s.  5 d.  I  far. 
as    12   pence    are    contained    times    in 

5609  pence,  or  467  shillings  and  5  pence  over.  Finally,  since  in 
20  shillings  there  is  i  pound,  in  467  shillings  there  are  as  many 
pounds  as  20  shillings  are  contained  times  in  467  shillings,  or 
23  pounds  and  7  shillings  over.  Therefore,  in  22437  farthings  there 
are  £23,  7s.  sd.  i  far.     Hence,  the 

EuLE. — Divide  the  given  denomination  by  the  mimier  re- 
quired of  this  denomination  to  mahe  a  unit  of  the  next  higher. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached.  Tlie  last  quotient^ 
with  the  several  remainders  annexed,  ivill  he  the  ansiver. 

Note. — The  remainders,  it  should  be  observed,  are  the  same 
denomination  as  the  respective  dividends  from  which  they  arise. 

278.  Proof. — Eeduction  Ascending  and  Descending 
prove  each  other ;  for,  one  is  the  reverse  of  the  other. 

6.  In  2047  pence  how  many  pounds,  shilhngs,  and  pence  ? 

7.  In  614  farthings,  how  many  shillings,  pence,  etc.? 

8.  How  many  pounds,  shillings,  etc.,  in  39610  farthings  ? 

9.  An  importer  paid  £27,  13s.  8d.  duty  on  a  package  of 
English  ginghams,  which  was  4d.  a  yard:  how  many 
yards  did  the  package  contain  ? 

10.  A  railroad  company  employs  1000  men,  paying  each 
4s.  6d.  per  day :  what  is  the  daily  pay-roll  of  the  company  ? 

11.  Reduce  18  lbs.  6  ounces  troy,  to  pennyweights. 


277.  How  are  lower  denominations  reduced  to  higher?    273.  Proof?    Explain 

Ex.  5  from  the  blackboard. 


EEDUCTlOi^.  191 

12.  Reduce  32  lbs.  9  oz.  5  pwt.  to  pennyweights. 

13.  How  many  pounds  and  ounces  in  967  ounces  troy? 

14.  How  many  pounds,  etc.,  in  41250  grains? 

15.  How  many  rings,  each  weighing  3  pwt.,  can  bo 
made  of  i  lb.  10  oz.  9  pwt.  of  gold  ? 

16.  What  is  the  worth  of  a  silver  cup  weighing  10  oz. 
16  pwfc.,  at  i2-|  cents  a  pennyweight? 

17.  Eeduce  165  lbs.  13  oz.  Avoirdupois  to  ounces. 

18.  Reduce  210  tons  121  pounds  8  ounces  to  ounces. 

19.  In  4725  lbs.  how  many  tons  and  pounds? 

20.  In  370268  ounces,  how  many  tons,  etc.? 

21.  What  will  875  lbs.  1 1  oz.  of  snuff  come  to,  at  5  cents 
an  ounce  ? 

22.  A  grocer  bought  i  ton  of  maple  sugar,  at  10  cents 
a  pound,  and  sold  it  at  6  cents  a  cake,  each  cake  weighing 
4  oz. :  what  was  his  profit  ? 

23.  In  250  lbs.  6  oz.  Apothecaries'  weight,  how  many 
drams  ? 

24.  In  2165  scruples,  how  many  pounds  ? 

25.  How  many  feet  in  1250  rods,  4  yards,  and  2  feet? 
For  the  metliod  of  multiplying  by  5^  or  i6|,  see  Art.  165. 

Ans.  20639  feet. 

26.  How  many  feet  in  365  miles  78  rods  and  8  feet? 

27.  In  5242  feet,  how  many  rods,  yards  and  feet? 

Remark. — In  order  to  divide  the  second  •5)^242  ft. 

dividend  1747  yards  by  5K  the  number  jv                -,       ni 

of  yards  in  a  rod,  we  reduce  both  the  ^^\    '^'  ^ 
divisor    and    dividend   to  halves;    then 


divide  one  by  the  other.     Thus,  5^  =  11      I  03494  half  yds, 
halves  ;  and  1747=3494  halves ;  and  11  is    ^/^5.  317  r.  3^  yd.  I  ft. 
contained  in  3494,  317  times  and  7  over. 

But  the  remainder  7,  is  lialxies ;  for  the  dividend  was  halves;  and 
7  half  yards=3^-  yards.     (Art.  168,  a,  169.) 

2^.  Reduce  38265  feet  to  miles,  etc. 

29.  How  many  rods  in  461  leagues,  2  miles,  6  furlongs? 


192  REDUCTION. 

30.  Reduce  7  m.  6  fur.  23  r.  5  yds.  8  in.  to  inches,  and 
prove  the  operiition. 

31.  Reduce  i  m.  7  fur.  39  r.  5  yds.  i  ft.  6  in.  to  inches, 
and  prove  the  operation. 

^32.  If  a  farm  is  2  J  miles  in  circumference,  what  will  it 
cost  to  enclose  it  with  a  stone  wall,  at  $1.85  a  rod? 

^^.  At  6J  cents  a  mile,  what  will  be  the  cost  of  a  trip 
round  the  world,  allowing  it  to  be  8299.2  leagues  ? 

34.  How  many  eighths  of  a  yard  in  a  piece  of  cloth 
57  yards  long? 

35.  How  many  sixteenths  of  a  yard  in  163  yards  ? 
^6.  In  578  fourths,  how  many  yards? 

37.  In  1978  sixteenths,  how  many  yards? 

38.  How  many  vests  will  16J  yards  of  satin  make,  allow- 
ing J  yard  to  a  vest  ? 

39.  A  shopkeeper  paid  $2.50  for  18J  yards  ribbon,  and 
making  it  into  temperance  badges  of  ^  yard  each,  sold 
them  at  12  J  cents  apiece :  what  was  his  profit  ? 

40.  How  many  square  rods  in  43816  square  feet? 

41.  How  many  acres,  etc.,  in  25430  square  yards  ? 

42.  Reduce  160  acres,  25  sq.  rods,  8  sq.  ft.  to  sq.  feet. 

43.  Reduce  100  sq.  miles  to  square  rods? 

44.  A  man  having  64  acres  8  sq.  rods  of  land,  divided 
it  into  6  equal  pastures:  how  much  land  was  there  in 
each  pasture  ? 

45.  My  neighbor  bought  a  tract  of  land  containing  15J 
acres,  at  I500  an  acre,  and  dividing  it  into  building  lots 
of  20  square  rods  each,  sold  them  at  $250  apiece:  how 
much  did  he  make  ? 

46.  Reduce  85  cubic  yards,  10  cu.  feet  to  cu.  inches. 

47.  Reduce  250  cords  of  wood  to  cu.  feet. 

48.  Reduce  18265  cu.  inches  to  cu.  feet,  etc. 

49.  Reduce  8278  cu.  feet  to  cords. 

50.  Reduce  164  bu.  i  pk.  3  qts.  to  quarts. 

51.  Reduce  375  bu.  3  pks.  i  qt.  i  pi  to  pints. 


KEDUCTI02<f.  193 

52.  Reduce  184  chaldrons  17  bu.  to  bushels. 

53.  Ill  8147  quarts  how  many  bushels,  etc.  ? 

54.  A  seedsman  retailed  75  bu.  3  pk.  of  clover  seed  at 
17  cents  a  quart:  what  did  it  come  to  ? 

55.  A  lad  sold  2  bu.  i  pk.  3  qts.  of  chestnuts,  at  S^  centu 
a  pint:  how  much  did  he  get  for  them  ? 

56.  Ill  98  quarts  i  pt.  2  gills,  how  many  gills? 

57.  In  150  gals.  3  qts.,  how  many  quarts? 

58.  In  45  barrels,  10  gals.  3  qts.,  how  many  quarts? 

59.  How  many  gills  in  17  hhds.  10  gals.  3  qts.  ? 

60.  IIow  many  gallons,  etc.,  in  86673  pints? 

-  61.  How  many  bottles  holding  ij  pt.  each  are  required 
to  hold  a  barrel  of  wine  ? 

^  62.  What  will  a  hogshead  of  alcohol  come  to,  at  6^  cents 
a  gill? 

63.  Reduce  30  days  5  hours  42  min.  10  sec.  to  seconds. 

64.  Reduce  17  weeks  3  days  5  hr.  30  min.  to  minutes. 

65.  Reduce  a  solar  year  to  seconds. 

66.  Reduce  6034500  sec.  to  weeks,  etc. 

67.  Reduce  5603045  hours  to  common  years. 

68.  Reduce  10250300  minutes  to  leap  years. 

69.  An  artist  charged  me  75  cents  an  hour  for  copying 
a  picture;  it  took  him  27  days,  working  io|  hours  a  day: 
what  was  his  bill  ? 

70.  In  48561  seconds,  how  many  degrees? 

71.  In  65237  minutes,  how  many  signs? 

72.  In  237°,  40',  31",  how  many  seconds? 

73.  In  360°,  how  many  seconds? 

74.  How  many  dozen  in  1965  things? 

75.  How  many  eggs  in  125  dozen? 

76.  How  many  gross  in  looooo? 

77.  How  many  pens  in  65  gross? 

78.  How  many  pounds  in  3  score  and  7  pounds  ? 

79.  How  many  sheets  of  paper  in  75  quires? 

80.  How  many  quires  of  paper  in  loooo  sheets? 


104:  APPLICi^TlOKS     OF 


APPLICATIONS   OF  WEIGHTS   AND    MEASURES. 

279.  The  use  of  Weights  and  Measures  is  not 

confined  to  ordinary  trade.  They  have  other  and  ?m- 
portant  applications  to  the  farm,  the  garden,  artificers' 
work,  the  household,  etc. 

THE     FARM     AND     GARDEN. 

280.  To  find  the  Contents  of  Rectangular  Surfaces. 

1.  How  many  square  yards  in  a  strawberry  bed,  4  yards 
long,  and  3  yards  wide  ? 

4  yards. 
Illustration. — Let  the  bed  be  rep- 

ressnted  by  the  adjoining'  figure ;  its 
length  being  divided  into  four  equal 
parts,  and  its  breadth  into  three,  each 
denoting  liiuar  yards.  The  bed,  ob- 
viously, contains  as  many  square  yards 
as  there  are  squares  in  the  figure. 
Now  as  there  are  4  squares  in  i  row,  in 
3  rows  there  must  be  3  times  4  or  12 
squares.     Therefore,  the  bed  contains  12  sq.  yards.     Henoa,  the 

EuLE. — Multijjly  the  length  hy  the  Ireadth.     (Art.  243.) 

Notes. — i.  Both  dimensions  should  be  reduced  to  the  same 
denomination  before  they  are  multiplied. 

2.  The  area  and  one  side  of  a  rectangular  surface  being  given,  the 
other  side  is  found  by  dividing  the  area  by  the  given  side. 

3.  One  line  is  said  to  be  multiplied  by  another,  when  the  number 
of  units  in  the  former  are  taken  as  many  times  as  there  are  like  units 
in  the  latter.     (Art.  47,  n.) 

2.  How  many  acres  in  afield  40  rd.  long  and  31  rd.  wide  ? 

wide  ? 

3.  If  the  width  of  an  asparagus  bed  is  66  feet,  what 
must  be  the  length  to  contain  J  of  an  acre  *? 

279.  What  13  paid  of  the  applications  of  Weights  and  Measures  ?  280.  How 
find  the  area  of  a  rectangular  surface  ?  Note.  If  the  dimensions  are  in  diflerent 
denonihiations,  what  is  to  be  done  ?    If  the  area  and  one  side  are  given  ? 


WEIGHTS     AND     MEASUKES.  195 

4.  What  is  the  length  of  a  pasture,  containing  1 5  acres, 
whose  width  is  :^si  rods  ? 

5.  A  speculator  bought  30  acres  of  land  at  $50  per 
acre,  and  sold  it  in  villa  lots  of  5  rods  by  4  rods,  at  $200  a 
lot:  what  was  his  profit? 

6.  A  garden  230  ft.  long  and  125  ft.  wide,  has  a  gravel 
walk  6  ft.  in  breadth  extending  around  it:  how  much 
land  does  the  walk  contain  ? 

7.  What  is  the  difference  between  2  feet  square  and 
2  square  feet  ? 

8.  A  farmer  having  3  acres  of  potatoes,  sold  them  at 
25  cents  a  bushel  in  the  ground ;  allowing  a  yield  of  2  J  bd. 
to  a  sq.  rod,  how  many  bushels  did  the  field  produce ;  and 
what  did  he  receive  for  them  ? 

281.    To  find  the   number  of  Hills,   Vines,   etc.,    in   a   given 
field,  the  Area  occupied  by  each  being  given. 

9.  How  many  hills  of  corn  can  a  farmer  plant  on  2  acres 
of  ground,  the  hills  being  4  ft.  apart  ? 

Analysis. — Since  the  hills  are  4  ft.  apart,  each  hill  will  occupy 
4  X  4  or  16  sq.  ft.  Now  2  acres=i6o  sq.  rods  x  272.25  x  2  =  87120  sq.  ft. 
Therefore,  he  can  plant  as  many  hills  as  16  is  contained  times  iu 
87120;  and  87120-^16=5445  hills.     Hence,  the 

Rule. — Divide  the  area  planted  hy  the  area  occupied  hy 
each  hill  or  vine. 

10.  How  many  bulbs  will  a  lady  require  for  a  crocus 
bed,  1 2  ft.  long  and  4  ft.  wide,  if  planted  6  inches  apart  ? 

11.  What  number  of  grape  vines  will  be  required  for 
f  of  an  acre,  if  they  are  set  3  ft.  apart  ?  What  will  it  cost 
to  set  them  at  $5.25  per  hundred  ? 

12.  How  large  an  orchard  shall  I  require  for  100  apple 
trees,  allowing  them  to  stand  33  ft.  apart  ? 

?8i.  Hew  find  the  number  of  hills,  vines,  etc.,  iu  a  given  field?  2S2.  How  a«N* 
ttooring,  i)]a3teriiig,  etc.,  estimated? 


196  APPLICATIOi^^S     OF 

ARTIFICERS'    WORK. 

282.  Flooring,  inlastering,  'painting,  papering,  roofing, 
paving,  etc.,  are  estimated  by  the  number  of  sq.ft.  or  sq.  yds. 
in  the  area,  or  by  the  ^'  square^^  of  loo  sq.  feet. 

^13.  What  will  be  tlie  cost  of  flooring  a  room  18  ft.  long 
and  1 6  ft.  6  in.  wide,  at  1 8 J  cts.  a  sq.  foot  ? 

Solution. — 18  ft.  x  16^=297  sq.  ft. ;  and  i8|  cts.  x  297=  $55.68!. 

14.  If  a  school  house  is  60  ft.  long  and  45  ft.  wide,  what 
will  be  the  expense  of  the  flooring,  at  $1.08  per  sq.  yard  ? 

15.  What  will  it  cost  to  plaster  a  ceiling  18  ft.  6  in.  long, 
and  15  ft.  wide,  at  ^3.20  per  square  of  100  sq.  ft. 

c  16.  What  will  be  the  cost  of  flagging  a  side-walk  206  ft. 
long  and  9I  ft.  wide,  at  $1.85  per  sq.  yard  ? 

17.  What  will  be  the  expense  of  painting  a  roof  60  ft. 
long  and  25  feet  wide,  at  $1.50  per  "square"  ? 

18.  What  cost  the  cementing  of  a  cellar  65  ft.  by  25  ft, 
at  $.25  per  square  yard  ? 

MEASUREMENT   OF   LUMBER. 

283.  To  find  the   Contents   of  Boards,  the  length   and 

breadth  being  given. 

Def. — A  Standard  Board,  in  commerce,  is  i  in.  tliick.  Hence, 
A  Hoard  F'oot  is  i  foot  long,  i  foot  wide,  and  i  inch  thick. 
A  Cubic  Foot  is  12  board  feet.     Hence,  the 

Rule. — Multiply  the  length  in  feet  dy  the  width  in  inches, 
and  divide  the  product  hy  12;  the  result  tuill  be  board  feet. 

Notes, — i.  If  a  board  tapers  regularly,  multiply  by  the  mean 
width,  which  is  half  the  sum  of  the  two  ends. 

2.  Shingles  are  estimated  by  the  thousand,  or  bundle.  They  are 
commonly  18  in.  long,  and  average  4  in.  wide.  It  is  customary  to 
allow  1000  to  a  square  of  100  feet. 

3.  The  contents  of  boards,  timber,  etc.,  were  formerly  computed 
by  duodecimals,  which  divides  the  foot,  into  12  in.,  the  inch  into  12 
sec.,  etc. ;  but  this  method  is  seldom  used  at  the  present  day. 

283.  The  thickness  of  a  standard  board  ?  What  is  a  board  foot  ?  A  cubic 
foot  ?    How  find  the  contents  of  a  board  ?    If  the  board  ie  taperiiig,  how  ? 


WEIGHTS     A:5fD     MEASURES.  19? 

19.  If  a  board  is  lo  ft.  long,  and  i  ft.  3  in.  wide,  what 
are  its  contents,  board  measure  ? 

Solution. — 10  x  15  =  150;  and  150-^-12  =  12^  board  ft.,  Ans. 

20.  What  are  the  contents  of  a  board  13  ft.  long,  and 
I  ft.  5  in.  wide  ? 

Solution. — 13  x  17=221 ;  and  221-^12=18 1\-  board  ft. 

21.  Required  the  contents  of  a  board  14  ft  long,  i  ft, 
4  in.  wide ;  and  its  value  at  7^  cts.  a  foot  ? 

22.  Required  the  contents  of  a  tapering  board  15  ft 
long,  14  in.  wide  at  one  end,  and  6  in.  at  the  other. 

23.  Required  the  contents  of  a  stock  of  9  boards  14  fi 
long,  and  i  ft.  2  in.  wide. 

24.  The  sides  of  a  roof  are  45  ft.  long  and  20  ft.  wide; 
what  will  it-  cost  to  shingle  both  sides,  at  $15.45  per  M., 
allowing  1000  shingles  to  a  square  ? 

284.  To  find  the  Cubical  Contents  of  Rectangular  Bodies, 

Rem. — Bound  Tiniber,  as  masts,  etc.,  is  estimated  in  cubic  feet. 
Heicn  Timber,  as  beams,  etc.  either  in  hoard  or  cubic  feet. 
Sawed  Tirriber,  as  planks,  joists,  etc.,  in  hoard  feet. 

25.  IIow  many  culjic  feet  are  there  in  a  rectangular 
block  of  marble  4  ft.  long,  3  ft.  wide,  and  2  ft.  thick  ? 

4  feet. 

Illustration. — Let  the  block  be 
represented  by  the  adjoining  diagram  ; 
its  length  being  divided  into  foiir 
equal  parts,  its  breadth  three,  and  its 
thickness  into  two,  each  denoting  linear 
feet. 

In  the  upper  face  of  the  block,  there  are  3  times  4,  or  12  sq.  ft.  Now 
if  the  block  were  i  foot  thick,  it  would  evidently  contain  i  time  as 
many  cubic  feet  as  there  are  square  feet  in  its  upper  face  ;  and  i  time 
4  into  3  =  12  cubic  feet.  But  the  given  block  is  2  ft.  thick ;  therefore 
i.  contains  2  times  4  into  3=24  cu.  ft.     Hence,  the 

Rule. — Multiply  the-  length,  breadth,  and  thichness 
to ii ether.     (Art.  249.) 

2S4-  How  are  hewn  timber,  etc.,  es  imated  ?   How  joiets,  studs,  etc.  ?   285.  Ho\« 


,.e^-   ^     .-'"•■■'  .---      .--■     ^ 

■^\^  ,'"    /    y    ^m 

rr~\ —  ;     -[. — liji 

Si 

r'=' ' !"~  i  "^w 

pi 

tK.  •      .:    ■    ;  »    !i/^ 

108  APPLICATIONS    OF 

285.  To  find  the  Contents  of  Joists,  &c.,  2,3,4,  &c.,  in.  thick. 

Rule. — Multiply  the  width  hy  such  a  part  of  the  lengthy 
as  the  thickness  is  o/  12  ;  the  result  will  be  board  feet. 

Notes. — i.  The  approximate  contents  of  round  timber  or  logs  may 
be  found  by  multiplying  \  of  the  mean  circumference  by  itself,  and 
this  product  by  the  length. 

2.  Cubic  feet  are  reduced  to  Board  feet  by  multiplying  them  by  12. 
For,  I  foot  board  measure  is  12  inches  square,  and  i  in.  thick ;  there- 
fore, 12  such  feet  make  i  cubic  foot.     Hence, 

If  one  of  the  dimensions  is  inches,  and  the  other  two  are  feet,  the 
product  will  be  in  Board  feet. 

3.  To  estimate  hay. — 

A  t07i  of  hay  upon  a  scaflPold  measures  about  500  cu.  ft. ;  when  in 
a  mow,  400  cu.  ft. ;  and  in  well  settled  stacks,  10  cu.  yards. 

4.  To  find  the  weight  of  coal  in  bins. — 

A  ton  (2000  lbs.)  of  Lehigh  white  ash,  egg  size,  measures  34^  cu.  ft. 
A  ton  of  white  ash  Schuylkill,  "  "         35  cu.  ft. 

A  ton  of  pink,  gray  and  red  ash,  '^  "         36  cu.  ft. 

26.  How  many  cu.  feet  in  a  stick  of  hewn  timber  20  ft. 
long,  I  ft.  3  in.  wide,  i  ft.  4  in.  thick  ?        Ans.  $^l  cu.  ft. 

27.  How  many  board  feet  in  a  joist  18  ft.  long,  5  in. 
wide,  and  4  in.  thick  ?    How  many  cubic  feet  ? 

Solution. — 4  is  i  of  12  ;  and  ^  of  18  ft.  is  6  ft.  Now  5  x  6=30 
bd.  ft. ;  and  30-7-12  =  2!^  cu.  ft. 

2S.  What  cost  45  pieces  of  studding  11  ft.  long,  4  in. 
wide,  and  3  in.  thick,  at  3  cts.  a  board  foot  ? 

29.  At  45  cts.  a  cu.  foot,  what  will  be  the  cost  of  a  beam 
32  ft.  long,  I  ft.  6  in.  wide,  and  i  ft.  i  in.  thick  ? 

30.  What  is  the  worth  of  a  load  of  wood  8  ft.  long,  4  ft. 
wide,  and  5  ft.  high,  at  $3^  a  cord  ? 

31.  How  many  tons  of  white  ash,  e-s  Sch.  coal,  will  a  bin 
10  ft.  long,  8  ft.  wide,  and  8  ft.  high,  contain.    A^is.  i8f  T 


find  the  contents  of  a  rectangular  body  ?    ITbte.  How  find  the  contents  of  round 
timber  ?    When  the  contents  and  two  of  the  dimensions  are  given,  how  find  the 
other  dimension?     How  reduce  cubic  feet  of  timber  to  beard  feet?     Why 
If  one  dimension  is  inches,  and  the  other  two  ft.,  what  is  the  product? 


WEIGHTS     AN^D     MEASURES.  199 

32.  How  many  cords  of  wood  in  a  tree  60  feet  high, 
whose  mean  circumference  is  10  feet? 

^^.  How  many  tons  of  hay  in  a  mow  22  ft.  hy  20  ft., 
and  15  ft.  high  ? 

34.  At  $18.50  a  ton,  what  is  the  worth  of  a  scaffold  oi 
hay  30  ft.  long,  12  ft.  wide,  and  10  ft.  high? 

286.  Stone  masonry  is  commonly  estimated  hy  the 
perch  of  25  cubic  feet. 

Excavations  and  embankments  are  estimated  by  the 
cubic  yard.    In  removing  earth,  a  cu.  yard  is  called  a  load. 

Brickworh  is  generally  estimated  by  the  1000,  but 
sometimes  by  cubic  feet. 

Notes. — i.  A  perch  strictly  speakings  24I  cu.  feet  being  16^  ft. 
long,  i^  ft,  wide,  and  i  ft.  liigh. 

2.  The  average  size  of  bricks  is  8  in.  long,  4  in.  wide,  and  2  in.  thick. 

In  estimating  the  labor  of  brickwork  by  cu.  feet,  it  is  customary  to 
measure  the  length  of  each  wall  on  the  outside ;  no  allowance  is 
made  for  windows  and  doors  or  for  corners. 

In  finding  the  exact  number  of  bricks  in  a  building,  we  should 
make  a  deduction  for  the  windows,  doors,  and  corners ;  also  an  allow- 
ance of  -,^,i  of  the  solid  contents  for  the  space  occupied  by  the  mortar. 

35.  What  will  be  the  cost  of  digging  a  cellar  45  feet 
long,  24  feet  wide,  and  8  feet  deep,  at  35  cents  a  cu.  yard  ? 

36.  At  $5.25  a  perch  (25  cu.  fi?.),  what  cost  the  labor 
of  building  a  cellar  wall,  i  ft.  6  in.  thick,  and  8  ft.  high, 
the  cellar  being  22  by  45  feet? 

37.  How  many  bricks  of  average  size  will  it  take  to 
build  the  wall»  of  a  house  50  ft.  long,  35  ft.  wide,  21  ft. 
high,  and  i  ft.  thick,  deducting  -^q  of  the  contents  for  the 
space  occupied  by  the  mortar,  but  making  no  allowance 
for  windows,  doors,  or  corners  ? 

38.  At  45  cents  a  cubic  yard,  what  will  it  cost  to  fill  in 
a  street  800  ft.  long,  60  ft.  wide,  and  4J  ft.  below  grade  ? 

2S6.  How  are  gtone  masonry,  excavationt,  etc.,  jictii^iaied ?  How  brick  work? 
Note.  The  average  size  of  bricks  ? 


200  A  P  P  L  I  C  A  T  I  O  K  S     OF 


THE    HOUSEHOLD. 


287.  To  find  the  Quantity  of  material  of  given  width  required 
to  line  or  cover  a  given  Surface. 

39.  How  many  yards  of  serge  |  yd.  wide  are  required  to 
line  a  cloak  containing  7|  yds.  of  camlet  i  J  yd.  wide  ? 

Analysis. — The  surface  to  be  lined=7|  yds.  x  ii=V^  x  f — ^r^  sq, 
yards  ;  i  yd.  of  the  lining=i  yd.  x  f =1  sq.  yd.  Now  as  |  sq.  yard 
of  camlet  requires  i  yard  of  lining,  the  whole  cloak  will  require  as 
many  yards  of  lining  as  f  is  contained  times  in  \^^ ;  and  \^^-v-^= 
\^^  X  f  =^/  or  15^  yards,  the  serge  required.     Hence,  the 

EuLE. — Divide  the  7iumher  of  square  yards  or  feet  in  the 
given  surface  hy  the  number  of  square  yards  or  feet  in  a 
yard  of  the  material  used. 

40.  How  many  yards  of  Brussels  carpeting  f  yd.  wide, 
are  required  for  a  parlor  floor  21  ft.  long  and  18  ft.  wide  ? 

41.  How  many  yards  of  matting  1 J  yd.  wide  will  it  take 
to  cover  a  hall  16  yds.  long  and  3^  yds.  wide  ? 

42.  How  much  cambric  ^  yd.  wide  is  required  to  line  a 
dress  containing  15  yds.  of  silk  |  yd.  wide  ? 

43.  How  many  sods,  each  being  15  in.  square,  will  be 
required  to  turf  a  court  yard  25  by  20  feet  ? 

44.  How  many  yards  of  silk  f  yd.  wide  are  required  to 
line  2  sets  of  curtains,,  each  set  containing  10  yds.  of 
brocatelle  i-J  yd.  wide? 

45.  How  many  marble  tiles  9  in.  square,  will  it  take  to 
cover  a  hall  floor  48  ft.  long  and  7^  ft.  wide  ? 

46.  How  many  rolls  of  wall  paper  9  yds.  long  and  i|  ft. 
wide,  are  required  to  cover  the  4  sides  of  a  room  18  by  16  ft. 
and  9  ft.  high,  deducting  81  sq.  ft.  for  windows  and  doors? 
'1  47.  How  many  gallons  of  water  in  a  rectangular  cistern, 
whose  length  is  8  ft.,  breadth  6  ft.,  and  depth  5  ft.  ? 

Note. — Guhic  measure  is  changed  to  gallons  or  bushels,  by  reducing 
the  former  to  cu.  inches,  and  duiding  the  result  by  231,  or  2150.4,  as 
the  case  may  be.     (Arts.  252,  255.) 

aSy.  How  fincl  the  quantity  of  material  required  to  line  or  cover  a  given  surface  f 


WEIGHTS     AKD     MEASUREb.  £01 

48.  A  man  constructed  a  cistern  containing  20  hogs- 
heads, the  base  being  6  ft-  square :  what  was  its  depth  ? 

49.  How  many  bushels  of  wheat  will  a  bin  6  ft.  long, 
\  ft.  wide,  and  3  ft.  deep,  contain  ? 

50.  How  many  cu.  feet  in  a  vat  containing  50  hogsheads? 

Note. — Gallons  and  bushels  are  reduced  to  cubic  inches  by  multi- 
plying them  by  231  or  21504,  as  the  case  may  be. 

51.  A  man  having  10  bu.  of  grain,  wishes  to  store  it  in 
a  box  5  ft.  long  and  3  ft.  wide :  what  must  be  its  height  ? 

52.  A  man  built  a  cistern  in  his  attic,  5  ft  long,  4  ft. 
wide,  and  3  ft.  deep :  what  weight  of  water  will  it  contain, 
allowing  1000  oz.  to  a  cu.  ft.  ? 

288.    To  change  Avoirdupois   Weight  to  Troy,  and  Troy 
Weight  to  Avoirdupois. 

53.  If  the  internal  revenue  tax  is  5  cents  per  ounce 
Troy,  on  silver  plate  exceeding  40  ounces,  what  will  be  the 
tax  on  plate  weighing  7  lb.  8  oz.  Avoirdupois  ? 

Analysis. — 7  lb.  8  oz.  Avoirdupois =120  oz. ;  and  120  oz.  x  437^  = 
52500  grains.  Now  52500  gr.-^ 480=109!  oz.  Troy.  Again,  109I  oz. 
—40  oz.  (exempt)=69l  oz. ;  and  5  cts.  x  69!= $3.46!.  Ans. 

54.  How  many  spoons,  each  weighing  i  oz.  Avoirdupois, 
can  be  made  from  i  lb.  9  oz.  17  pwt.  12  gr.  Troy? 

Analysis. — i  lb.  9  oz.  17  pwt.  12  gr.  =  10500  grains;  and  1050Q 
gr8.-r-437.5  grs.  (i  oz.  Avoir.)=24  spoons.  Ans.    Hence,  the 

EuLE. — JRediice  tlie  given  quantity  to  grains;  then  reduce 
the  grains  to  the  denominations  required.     (Art.  231.) 

55.  If  a  family  have  1 2  lb.  6  oz.  Avoir,  of  silver  plate,  what 
will  be  the  tax,  exempting  40  oz.,  at  5  cts.  per  oz.  Troy  ? 

56.  A  lady  bought  a  silver  set  weighing  11  lbs.  4  oz. 
Troy  :  how  many  pounds  Avoir,  should  it  weigh  ? 

57.  How  many  rings,  each  weighing  3 J  pwt.,  can  be 
made  from  a  bar  of  gold  weighing  i  pound  Avoirdupois  ? 

58.  What  is  the  value  of  a  silver  pitcher  weighing  2  lb. 
8  oz.  Avoir.,  at  $2  per  ounce  Troy? 

59.  A  miner  sold  a  nugget  of  gold  weighing  3 J  lbs. 
Avoir.,  at  $17  per  oz.  Troy:  what  did  he  get  for  it? 


DENOMINATE    FRACTIONS, 

289.  A  Denomhiate  JFractioii  is  one  or  more  of 
the  equal  parts  into  which  a  compou?id  or  denominate 
number  is  divided. 

Denominate  Fractions  are  expressed  either  as 
common  fractions,  or  as  decimals.  The  former  are  usually 
termed  denominate  fractions;  the  latter,  denominate 
decimals. 

REDUCTION    OF    DENOMINATE   FRACTIONS. 

CASE    I. 

290.    To   reduce   a    Denominate    Fraction   from   Higher 

denominations  to  Loiver. 

1.  What  part  of  an  inch  is  -f^  of  a  yard  ? 

Analysis. — ^  of  i  yard  equals  -J-^  of  2  yards.  2  vd.  X  3  =  6  ft. 
■Reducing  the  numerator  to  inches,  we  have  6  ft.  X  12  — 72  in. 
2  yards=2  x  3  x  12,  or  72  inches ;  and  g^^  of  72  ji^is,  '2  or  ^  in. 
in.  is  ^1  or  f  inch.  ^ 

Or,  denoting  the  multiplications,  and  cancelling  the  factors 
common     to     the     numerators      and     denominators,    we     have 

2      ,      2x3  „^      2x3x12.        2x3x113. 

-_  yd.=— ^  ft- ^- m.= ^ =^  in.     Hence,  the 

96  96  96  9S,  4         4 

KuLE. — Reduce  the  numerator  to  the  denomination  re- 
quired, and  place  it  over  the  given  denominator;  cancelling 
the  factors  common  to  hoth.     (Art.  276.) 

Note. — The  steps  in  this  operation  are  the  same  as  those  in 
Ueduction  Descending. 

2.  Eeduce  ^  of  a  bushel  to  the  fraction  of  a  quart. 

3.  Reduce  £^z  to  the  fraction  of  a  penny. 

4.  Reduce  y^nnr  of  a  day  to  the  fraction  of  an  hour. 

5.  What  part  of  an  ounce  is  -^  of  a  pound  ? 

6.  What  part  of  a  square  inch  is  -j^  of  a  sq.  foot  ? 

289.  What,  arc  denominate  fractions?  How  are  they  expressed?  200.  How 
ftr«  denominate  fractions  reduced  from  higher  denominations  to  lower?  Note 
WTiat  are  the  steps  in  this  operation  like  ?    Explain  Ex.  i  from  the  hlackboard. 


de:n"o  MI  Innate    fractloks. 

CASE    II. 

291.    To   reduce  a   Denominate   Fraction  from   Lower 
denominations  to  Higher, 

7.  What  part  of  a  yard  is  f  of  an  iucli  ? 

Analysis. — Reducing  i  yard  to  fourths  of  an  inch,  we  have  i  jd. 
<=36  in. ;  and  36  in.  x  4  =  144  fourths  inch.  Now  3  fourths  are  7^4  ot 
£44  fourths  ;  therefore  f  of  an  inch  is  yf  4  or  /g  of  a  yard.    Hence,  the 

Rule. — Reduce  a  unit  of  the  denomination  in  which  the 
required  fraction  is  to  he  ex23ressed,to  the  same  denominatfion 
as  the  given  fraction,  and  place  the  numerator  over  it. 

8.  What  part  of  a  dollar  is  f  of  a  mill  ? 

9.  What  part  of  a  pound  is  |-  of  a  farthing  ? 
10.  What  part  of  a  Troy  ounce  is  ^  of  a  grain  ? 

j"  II.  Reduce  f  of  a  gill  to  the  fraction  of  a  gallon. 

12.  Reduce  f  of  a  rod  to  the  fraction  of  a  mile. 

13.  Reduce  -J  of  a  pound  to  the  fraction  of  a  ton. 

CASE   III. 

292.    To    reduce    a   Deuoniinate  Fraction  from    higher 
to  a   Whole  Naniher  of  lower  denominations. 

14.  Reduce  f  of  a  yard  to  feet  and  inches. 

Analysis. — Reducing  the  given  fraction  to  the  next  lower  denomi- 
nation, we  have  f  yard  x  3=:V^  ft.  or  2  ft.  and  %  ft.  remainder. 

Again,  reducing  the  remainder  to  the  next  lower  denomination, 
W3  have  |  ft.  x  12  =  2.^'*-,  or  4*  in.  Therefore,  |  yards  equals  2  ft.  4!^  in. 
(Arts.  147,  276.)     Hence,  the 

Rule. — MaUiply  the  given  numerator  hy  the  number 
required  to  reduce  the  fraction  to  the  next  lower  denomiTia- 
tion,  and  divide  the  product  hy  the  denominator.  (Art.  2^6.) 

Midtiply  and  divide  the  successive  remainders  in  the 
same  manner  till  the  loivest  denomination  is  reached.  The 
several  quotients  will  he  the  answer  required. 

2-)i.  How  are  denominate  fractions  reduced  from  lower  denominations  to 
liigher?  202.  How  are  denominate  fractions  reduced  from  lii^her  to  whole 
nnmbers  of  lower  denominatioud  ? 


20-1  DENOMINATE     FRACTIONS. 

15.  Eeduce  J  of  an  eagle  to  dollars  and  cents.  A?is.  $7.50 

16.  Eeduce  f  of  a  pound  sterling  to  shillings  and  pence 

17.  How  many  pecks  and  quarts  in  -f-|  of  a  bushel  ? 

18.  How  many  pounds  in  |  of  a  ton  ? 

19.  How  many  ounces,  etc.,  in  |^  of  a  Troy  pound  ? 

20.  What  is  the  value  of  y\  of  a  mile  ? 

21.  "What  is  the  value  of  H  of  an  acre  ? 

22.  Eeduce  |^  of  a  cord  to  cu.  feet. 

CASE    IV. 

293.    To  reduce  a   Compound  Number  from   lower  to  a 
Denominate  Fraction  of  higher  denominations. 

23.  Eeduce  2  ft.  3  in.  to  the  fraction  of  a  yard. 
Analysis. — 2  ft.  3  in.=27  inches;  and  i  yard=36  inches.     The 

question  now  is,  what  part  of  36  in.  is  27  in.  ?    But  27  is  f  ^  or  f  of  36. 
(Art.  173.)     Therefore  2  ft.  3  in.  is  f  of  a  yard.     Hence,  the 

EuLE. — I.  Reduce  the  given  number  to  its  lowest  denom- 
ination for  the  numerator. 

II.  Reduce  to  the  same  denomination,  a  U7iit  of  the 
required  fraction,  for  the  denominator. 

Notes. — i.  If  the  lowest  denomination  of  the  ^iven  number  con- 
tains a  fraction,  the  number  must  be  reduced  to  the  parts  indicated 
by  the  denominator  of  the  fraction. 

2.  If  it  is  required  to  find  lohat  part  one  compound  number  is  of 
another  which  contains  different  denominations,  reduce  hoth  to  thi 
lowest  denomination  mentioned  in  either,  and  proceed  as  above. 

S\  24.  Eeduce  2  qt.  i  pt.  3I  gills  to  the  fraction  of  a  gal.  ? 

25.  What  part  of  a  pound  Troy  is  5  oz.  2  pwt  10  gr.  ? 

26.  Eeduce  3  pk.  2  qt.  i  pt.  to  the  fraction  of  a  husheL 

27.  Eeduce  6  cwt.  48  lb.  to  the  fraction  of  a  ton. 

28.  What  part  of  a  square  yard  is  5  sq.  ft.  2 of  sq.  inches  ? 

29.  What  part  of  £3,  los.  2d.  is  7s.  gd.  2  far.  ? 

30.  What  part  of  a  week  is  3  days,  5  hrs.  40  min  ? 

31.  What  part  of  a  cord  is  2  5f  cu.  feet  of  wood  ? 

32.  What  part  of  5  miles  2  fur.  14  rods  is  2  m.  i  fur.  2  rods  ? 

203.  How  arc  denominate  numbers  reduced  from  lower  denoml  aations  ta 
fractions  of  a  hij^hicr  denomiuation  V    Explaiu  Ex.  2-  •'"rem  the  Islaclcbc  ard. 


DE^-OMINATB     DECIMALS.  ^05 


CASE    V. 

294.   To  reduce  a  JDenonhimate  Decimal  from  a  higher  to  a 
Compound  Xumhei-  of  lower  denominations. 

^2,'  Reduce  .89375  gallon  to  quarts,  pints,  and  gills? 

Analysis. — This  example  is  a  case  of 
Reduction  Descending.     (Art.  276.)     We  -89375  g^l- 

therefore  multiply  the  given  decimal  of  a  4 

gallon  by  4  to  reduce  it  to  the  nest  lower  3'57500  ^t. 

denomination,  and  pointing  off  the  pro-  2 

duct  aa  in  multiplication  of  decimals,  the  > 

result  is  3  qts.  and  .57500  qt.     In  like  "  ' 

manner,  we  multiply  the  decimal  .57500  —   . 

qt.  by  2  to  reduce  it  to   pints,  and  the  .60006  gl. 

result  is  i  pt.  and  .15000  pt.     Finally,  we     Ans.  3  qt.  i  pt.  0.6  ^i 
multiply  the  decimal  .15000  pt.  by  4  to 

reduce  it  to  gills,  and  the  result  is  .6  gill.     Therefore,  .89375  gal. 
equals  3  qt.  i  pt.  0.6  gi.     Hence,  the 

Rule. — Multiply  the  denominate  decimal  ly  the  mi^nher 
required  of  the  next  loiver  denomi7iation  to  make  one  of  the 
given  denomination^  and  point  off  the  product  as  in  multi^ 
plication  of  decimals.     (Arts.  191,  276.) 

Proceed  in  this  manner  ivith  the  decimal  part  of  the  suc- 
cessive ^products,  as  far  as  required.  The  integral  part  of 
the  several  products  ivill  he  the  a^iswer. 

34.  What  is  the  value  of  £.125445  in  shillings,  etc.  ? 

35.  "What  is  the  value  of  .91225  of  a  Troy  pound  ? 

36.  Reduce  .35  mile  to  rods,  etc. 

37.  How  many  quarts  and  pints  in  .625  of  a  gallon  ? 
2,2>.  How  many  minutes  and  seconds  in  .651  degree? 

/    39.  How  many  days,  hours,  etc.,  in  .241  week  ? 

4c.  What  is  the  value  of  .25256  ton  ? 

41.  What  is  the  value  of  .003  of  a  Troy  pound? 

42.  What  is  the  value  of  £5.62542  ? 


204.  How  are  denominate  decimals  rednred  from  higher  denominations  to 
whole  numbers  of  a  lower  denomination  ?    Explain  Ex.  33  upon  the  blackbcfiro 


206  DEK0  3IIKATE     DECIMALS. 


CASE    VI. 

295.    To   reduce  a  Compound  Number  from  lower  to  a 
Doiiominate  Decimal  of  a  higher  denomination. 

43.  Eeduce  3  pk.  2  qt.  i  j)t.  to  the  decimal  of  a  bushel 
Analysis. — 2  pints  make  i  quart ;  hence  operation. 

ipt. 


2^t. 

3-3125  pk. 


tliere  is  \  as  many  quarts  as  pints,  and  i  of  2 

I  qt.  is  .5  qt.,  wliicli  we  write  as  a  decimal  on  o 

the  right  of  the  given  quarts.    In  like  manner 

there  is  \  as  many  pecks  as  quarts,  and  \  of  4 

2.5  is  .3125  pk.,  which  we  place  on  the  right     Ans.  .8281*25  bu. 

of  the  given   pecks.     Finally,  there  is  |  as 

many  bushels  as  pecks,  and  \  of  3.3125  is  .828125  bu.     Therefore, 

3  pk.  2  qt.  I  pt.  equals  .828125  bushel.     Hence,  the 

EuLE. —  Write  the  numbers  in  a  column,  inlacing  the 
lowest  denomination  at  the  top. 

Beginning  with  the  lowest,  divide  it  hy  the  numler  re- 
quired of  this  denomination  to  make  a  unit  of  the  next 
higher,  and  annex  the  quotient  to  the  next  denomination. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached. 

44.  Eeduce  6  oz.  10  pwt.  4  gr.  to  the  decimal  of  a  pound, 

45.  Eeduce  10  lb.  to  the  decimal  of  a  ton. 

46.  Eeduce  3  fur.  25  r.  4  yd.  to  the  decimal  of  a  mile. 

47.  Eeduce  9.6  pwt.  to  the  decimal  of  a  Troy  pound. 
What  decimal  part  of  a  barrel  are  15  gal.  3  qt. 

49.  What  decimal  part  of  2  rods  are  2\  fathoms  ? 

50.  What  decimal  part  of  £3,  3s.  6d.  are  15s.  io.5d. ; 

51.  What  part  of  a  barrel  are  94.08  lbs.  of  flour? 

52.  Eeduce  7.92  yds.  to  the  decimal  of  a  rod. 

53.  Eeduce  i  day  4  hr.  10  sec.  to  the  decimal  of  a  w< 

54.  Eeduce  45  sq.  rods  25  sq.  ft.  to  the  decimal  of  an  a 

55.  Eeduce  53!^  cu.  feet  to  the  decimal  of  a  cord. 


\)^ 


295.  How  are  compound  numbers  reduced  from  lower  denominationa 
denominate  decimals  of  a  higher?    Explain  Ex.  43  upon  the  blackboard? 


METEIO    WEIGHTS    AND 
MEASURES, 

296.  Metric  Weights  and  Pleasures  are  founded 
upon  the  decimal  notation,  and  are  so  called  because  their 
primary  unit  or  base  is  the  3Ieier.* 

297.  The  Meter  is  the  miit  of  le^igth,  and  is  equal  to 
one  ten-millionth  part  of  the  distance  on  the  earth's  surface 
from  the  equator  to  the  pole,  or  39.37  inches  nearly. 

Notes. — i.  The  term  metei'  is  from  the  Greek  metron,  a  measure. 
2.  The  standard  meter  is  a  har  of  platinum  deposited  in  the 
archives  of  Paris. 

298.  The  Metric  System  employs /z^e  different  units  or 
denominations  and  seven  prefixes. 

The  units  are  the  me'ter,  U'ter,  gram,  ar,  and  ster.^ 

299.  The  names  of  the  higher  denominations  are  formed 
by  prefixing  to  the  unit  the  Greek  numerals,  deJc'a  10, 
hek'to  100,  JciTo  1000,  and  myr'ia  loooo ;  as,  deh'a-me'ter, 
10  meters;  hek'to-me'ter,  100  meters,  etc. 

The  names  of  the  lower  denominations,  or  divisions 
of  the  unit,  are  formed  by  prefixing  to  the  unit  the  Latin 
numerals,  dec'i  (des'-ee)  .1,  cen'ti  (cent'-ee)  .01,  and  mil'li 
(mil'-lee)  .001;  as,  dec'i-me'ter,  y\j  meter;  cen'ti-me'ter, 
yj^  meter ;  mil'li-me'ter,  -j^^^  meter. 


Note. — The  numeral  prefixes  are  the  key  to  the  whole  system  ; 
their  meaning  therefore  should  be  thoroughly  understood. 

296.  Upon  what  are  the  Metric  Weights  and  Measures  founded  ?  Why  so 
called  ?  297.  What  is  the  meter  ?  299.  How  are  the  names  of  the  lower  denom- 
inations formed  ?    The  higher  ? 

*  This  system  had  its  origin  in  France,  near  the  close  of  the  last  century.  Its 
simplicity  and  comprehensiveness  have  secured  its  adoption  in  nearly  all  the 
countries  of  Europe  and  South  America.  Its  use  was  legalized  in  Great  Britain 
in  1864  ;  and  in  the  United  States,  by  Act  of  Congress,  1866. 

t  The  spelling,  pronunciation,  and  abbreviation  of  metric  terms  in  this  work, 
are  the  same  as  adopted  by  the  American  Metric  Bureau,  Boston,  and  the 
Metrological  Society,  New  York. 


lo  cen'ti-me'ters 

lo  decl-me  ters 

lo  meters 

lo  dek'a-me'ters 

lo  hek'to-me  ters 

10  kilo-meters 

208  METKIC     OR     DECIMAL 


LINEAR    MEASURE. 

300.  The  TJnit  of  Length  is  the  Meter,  whicli  is 
equal  to  39.37  inclies.* 

The  denominations  are  the  mil'li-rne'terf  cen'ti-rnetery 
dec'i-me'ter,  meter,  deh'a-meter,  hek' to-me' ter ,  kil'o-me'tery 
and  myria-mdter. 

TABLE. 
ID  mil'li-me'ters  {mm.)  make  i  cen'ti-me'ter  -    -    mi. 

decimeter  -  -  dm. 
3IE'TEll  -  -  m. 
dek'a-meter  -  -  Dm. 
hek'to-me'ter  -  -  Hm. 
Ml' o-me' ter  -  -  Km. 
myr'ia-me'ter     -    Mm. 

Notes. — i.  The  Accent  of  each  unit  and  jirejix  is  on  the  first 
syllable,  and  remains  so  in  the  compound  words. 

To  abbreviate  the  compounds,  pronounce  only  the  prefix  and  the 
first  letterof  the  unit ;  as,  centim,  millim,  centil,  decig,  hektug,  etc. 

2,  The  meter,  like  our  yard,  is  used  in  measuring  cloths,  laces, 
moderate  distances,  etc. 

For  long  distances  the  kilometer  (3280  ft.  10  in.)  is  used  ;  and 
for  rninute  measurements,  the  centimeter  or  millimeter. 

3.  Decimeters,  dekameters,  hektometers,  like  our  dimes  and  eagles, 
are  seldom  used. 

SQUARE    MEASURE. 

301.  The  Unit  for  measuring  surfaces  is  the  Square 
Meter,  which  is  equal  to  1550  sq.  in. 

The  denominations  are  the  sq.  cen'ti-yne'ter,  sq.  dec'i- 
me'ter,  and  sq.  me' ter. 

100  sq.  cen'tim.  make  i  sq.  dec'i-me'ter  -    -    sq.  dm. 
100  sq.  dec'im.        *'      i  Sq.  3Ie'ter      -    -    sq.  m. 

Note. — The  square  meter  is  used  in  measuring  floorings,  ceilings, 
etc. ;  square  deci-meters  and  centi-meters,  for  minute  surfaces. 

300.  What  is  the  unit  of  Linear  Measure  ?  Its  denominations  ?  Recite  the 
Table.    301.  What  is  the  unit  for  measuring  surfaces  ?    Recite  the  table. 

*  Authorizetl  by  Act  of  Con^rcs.^^,  1866. 


WEIGHTS     AND     MEASURES.  209 

302.  The  JJtiit  for  measuring  land  is  the  Ar,  which 
is  equal  to  a  square  dekameter,  or  119.6  sq.  yards. 

The  only  suMivision  of  the  Ar  is  the  cen'tar ;   and 

the  only  multiple  is  the  heJc'tar.     Thus, 

100  cent'ars  (ca.)  make  i  Ar    -    -    -    a. 
100  ars  "       I  hek'tar  -    -    Ha. 

Notes. — i.  The  term  ar  is  from  the  Latin  area,  a  surface. 

2.  In  Square  Measure,  it  takes  100  units  of  a  /oip^r  denomination 
to  make  one  in  the  next  higher  ;  hence  each  denomination  must  have 
two  places  of  figures.     In  this  respect  centars  correspond  to  cents. 

CUBIC    MEASURE. 

303.  The  Unit  for  measuring  solids  is  the  CuMc 
Meter,  which  is  equal  to  35.316  cu.  ft. 

TABLE. 

1000  cu.  mil'lim.     make  i  cu.  cen'ti  me'ter  -  cu.  cm. 
1000  cu.  cen'tim.        "      i  cu.  dec'i  me'ter    -  eu.  dm. 
1000  cu.  dec'im.         "      i  Cu.  Me'ter      -  cu.  m. 
Notes. — i.  The  cubic  meter  is  used  in  measuring  embankments, 
excavations,  etc. ;  cubic  centimeters  and  millimeters ^iov  minute  bodies. 

2.  Since  it  takes  1000  units  of  a  lower  denomination  in  cubic 
measure  to  make  one  of  the  next  higher,  it  is  plain  that,  like  mills, 
each  denomination  requires  three  places  of  figures. 

304.  In  measuring  wood,  the  Ster,  which  is  equal  to 
a  cuhic  meter,  is  sometimes  used. 

The  only  subdivision  of  the  ster  is  the  dec'i-ster  ;  and 
the  only  multiple,  the  deh'a-ster.     Thus, 

10  dec'i-sters  make  i  Ster  -    -    -    8. 
10  sters  "       I  dek'a-ster    -    Bs. 

Notes. — i.  The  term  ster  is  from  the  Greek  stereos,  a  solid. 
2.  In  France,  fireicood  is  commonly  measured  in  a  cubical  box, 
whose  length,  breadth,  and  height  are  each  i  meter. 

3.  As  the  ster  is  applied  only  to  wood,  and  probably  will  never 
come  into  general  use,  its  divisions  and  multiples  may  be  omitted. 

Note.  How  many  places  of  figure?  does  each  denomination  occupy  ?    Wl  y  ? 

302.  What  is  the  unit  for  measuring  Iniid?     Its  divisions?      Its  miil(i)]es? 

303.  What  are  the  units  for  measuring  solids  ?    Recite  the  Table.    How  many 
places  does  each  denomination  in  cubic  measure  occupy  ?    Why  ? 


210 


METRIC     OR     DECIMAL 


DRY    AND    LIQUID    MEASURE. 

305.  The  Unit  of  Dry  and  Liquid  3l€a8iire 

is  the  Li'ter  {lee'ter),  which  is  equal  to  a  cuMc  decim,  or 
t.0567  liquid  quart  or  0.908  dry  quart. 

The   denominations    are    the   mil'li-U'ter,   cen'ti-li'ter, 
dec'i-li'ter,    li'ter,    deh'a-li'ter,     hek'to-U'ter,    kil'o-liter. 


10  mil'li-li'ters  {mi)  make  i  cen'ti-li'ter 


10  cen'ti-li'ters 

10  dec'i-li'ters 

10  li'ters 

10  dek'a-li'ters 

10  hek  to-liter 

10  kilo li ters 

d. 

dL 

I. 

DL 

HI. 

KL 

ML 


dec'i-li'ter  - 
LVter  -  - 
dek'a-liter  - 
hek'to-li'ter 
kil'o-li'ter  - 
myria-li'ter 

Notes. — i.  The  liter  is  used  in  measuring  milk,  wine,  etc.  For 
small  quantities,  the  centiliter  and  milliliter  are  employed  ;  and  for 
large  quantities,  the  dekaliter. 

2.  For  measuring  grain,  etc.,  the  heJctoliter,  which  is  equal  to 
2.8375  bushels,  is  commonly  used. 

3.  The  term  liter  is  from  the  Greek  litra,  a  pound. 


WEIGHT. 

306.  The  Unit  of  Weight  is  the  Gram,  which  is 
equal  to  15.432  grains. 

The  denominations  .are  the  mil'li-gram,  ceji'ti-gram, 
dec'i-gram,  gram,  deh'a-gram,  Tiehto-gram,  kW o-gram, 
ynyr'ia-gram,  and  ton'neau  or  ton. 


10  milligrams  {mg.)  make  i  cen'ti-gram    - 


eg. 


10  centigrams 

I  decigram      -    - 

dg. 

10  dec'i-grams 

I  Gram      -    -    - 

g- 

10  grams 

I  dek'a-gram     -    - 

Bg. 

10  dek'a-grams 

1  hek'to-gram  -    - 

Hg. 

10  hek  to-grams 

I  KiUo-(fr<n}i   ■ 

Kg. 

10  kilo-grams 

I  myr'ia-gram  -    - 

Mg. 

100  myr'ia-grams 

I   TONNEAU  or  Ton 

T. 

305.  What  is  the  UTiU  of  Dry  and  Liquid  Measure  ?    Its  denominations  ?    Re- 
peat, the  Table.    306.  What  is  the  unit  of  Weight  ?    Its  denominations  ? 


WEIGHTS     AJ^D     MEASURES.  211 

Remark.— As  the  quintal  (lo  Mg.)  is  seldom  used,  and  millier 
means  the  same  as  tonneau  or  ton,  both  these  terms  may  be  dropped 
from  the  table.     The  metric  ton  is  equal  to  2204.6  lbs. 

Notes. — i.  This  Table  is  used  in  computing  the  weight  of  all 
objects,  from  the  minutest  atom  to  the  largest  heavenly  body. 

2.  The  gram  is  derived  from  the  Greek  gramma.,  a  rule  or 
standard,  and  is  equal  to  a  cvhic  centimeter  of  distilled  water  at  its 
greatest  density,  viz.,  at  the  temperature  of  4''  centigrade  thermome- 
ter, or  39.83°  Fahrenheit,  weighed  in  a  vacuum. 

3.  The  common  unit  for  weighing  groceries  and  coarse  articles 
is  the  kilogram,  which  is  equal  to  2.2046  pounds  Avoirdupois. 

4.  Kilogram  is  often  contracted  into  kilo,  and  tonneau  into  ton. 

307.  To  express  Metric  Denominations  decimally  in  terms 
of  a  given  Unit. 

T.  Write  7  Hm.  o  Dm.  9  m.  3  dm.  5  cm.  decimally  in 
terms  of  a  meter. 

Analysis. — Metric    denominations    in-  operation. 

crease  and  decrease  by  the  scale  of  10,  and  7°9-35  ^v  -4/25. 

correspond  to  the  orders  of    the   Arabic 

Notation.  We  therefore  write  the  given  meters  as  units,  the  deka- 
meters  as  tens,  and  the  hektometers  as  hundreds,  with  the  deci- 
meters and  centimeters  on  the  right  as  decimals.    Hence,  the 

Rule. —  Write  tlie  denominations  alove  the  given  unit 
in  their  order,  on  the  left  of  a  decimal  point,  and  those 
beloiu  the  unit,  on  the  right  as  decimals. 

Notes. — i.  If  any  intervening  denominations  are  omitted  in  the 
given  number,  their  places  must  be  supplied  by  ciphers. 

2.  As  each  denomination  in  square  measure  occupies  two  places  of 
figures,  in  writing  square  decimeters,  etc.,  as  decimals,  if  the  number 
is  less  than  10,  a  cipher  must  be  prefixed  to  the  figure  denoting  them. 

Thus,  17  sq.  m.,  and  2  sq.  dm.  =  17.02  sq.  meters.     (Art.  302,  ri.) 

3.  In  like  manner,  in  writing  cubic  decimeters,  etc.,  as  decimals, 
if  the  number  is  less  than  10,  two  ciphers  must  be  prefixed  to  it. 

Thus,  63  cu.  m.  and  5  cu  dm.  =  63.005  cu.  meters. 


Repeat  the  Table.  Note.  The  common  unit  for  weighing  groceries,  etc.  ? 
307.  IIow  write  metric  quantities  in  terms  of  a  given  unit  ?  Note.  If  any  de- 
nomination is  omitted,  what  is  to  be  done  ? 


212  METRIC     OR     DECIMAL 

308.  Metric  denominations  expressed  decundlly  are  read 
like  numbers  in  tlie  Arabic  notation. 

Thus,  the  Answer  to  Ex.  i,  (709.35),  is  read,  "seven  hundred 
and  nine,  and  thirty-five  hundredths  meters,  or  seven  hundred 
nine  meters,  thirty-five,"  as  we  read  $709.35,  "seven  hundred 
nine  dollars,  thirty -five,"  omitting  the  name  cents. 

Note. — The  number  of  figures  following  the  name  of  the  unit, 
shows  the  denomination  of  the  last  decimal  figure. 

If  the  name  is  required,  it  is  better  to  use  abbreviations  of  the 
metric  terms,  as  centims,  millims,  etc.,  than  the  English  words 
hundredths,  or  thousandths  of  a  meter,  etc.  The  former  are  not  only 
significant  of  the  kind  of  unit  and  the  value  of  the  decimal,  but 
are  understood  by  all  nations. 

Write  decimally,  and  read  the  following : 

1.  5  Mm.,  3  Km.,  7  Hm.,  c^Dm.,  9  m.,  8  dm.,  4  cm., 
5  mm.  in  terms  of  a  meter,  hektometer,  kilometer,  and 
decimeter. 

2.  4  Kg.,  5  Hg.,  o  Dg.,  5  g.,  I  dc,  o  eg.,  8  mg.  in  terms 
of  a  dekagram,  centigram,  hektogram,  and  kilogram. 

3.  Write  and  read  5  KL,  o  HI.,  6  DL,  3  1.,  o  dl.  8  cl. 
in  terms  of  a  hektoliter.  Ans.  50.6308  hektoliters. 

309.  To  reduce  higher  Metric  Denominations  to  lower. 

Ex.  I.  Reduce  46  3275  kilometers  to  meters. 

Analysis  —From  kilometers  to  meters  operation. 

there    are    three   denominations,    and  it  A^'Z^IS  Km. 

takes  10  of  a  lower  denomination  to  make  ^QQQ 

a  unit  of  the  next  higher.     We  therefore     Ans.  46327.5000  m. 

multiply  by  1000,  or  remove  the  decimal 

point  three  places  to  the  right.     (Art.  181.)    Hence,  the 

Rule. — Multiply  the  given  denomination  ly  10,  100, 
1000,  etc.,  as  the  case  may  require  ;  and  point  off  tlie  prod- 
uct as  in  midti'plication  of  decimals.     (Art.  191.) 

Note. — It  should  be  remembered  that  in  the  Metric  System  each 
denomination  of  square,  measure  requires  two  figures;  and  each 
denomination  of  cnhic  measure,  tlLrte  figures. 


WEIGHTS    a:sd    measures.  313 

g.   In  43.75  hectares,  how  many  square  meters? 

3.  Reduce  867  kilograms  to  grams. 

4.  Reduce  264.42  hectoliters  to  liters. 

5.  In  2561  ares,  how  many  square  meters? 

6.  In  8652  cubic  meters,  how  many  cubic  decimeters? 

7.  Reduce  4256.25  kilograms  to  grams. 

310.  To  reduce  lower  Metric  Denominations  to  higher. 

8.  Reduce  84526.3  meters  to  kilometers. 

Analysis. — Since  it  takes  10  linear  units 
to  make  one  of  the  next  higher  denomination,  otora-tion. 

it  tollovvs  that  to  reduce  a  number  from  a      100QJ84520.3  I"- 
lower  to  the  next   higher   denomination,  it  84.5263  km. 

must  be  divided  by  10 ;  to  reduce  it  to  the 

next  higher  still,  it  must  be  again  divided  by  10,  and  so  on.  From 
meters  to  kilometers  there  are  three  denominations ;  we  therefore 
divide  by  1000,  or  remove  the  decimal  point  three  places  to  the  left. 
The  answer  is  84.5263  km.    Hence,  the 

Rule. — Divide  the  given  denomination  hy  10,  100,  1000, 
etc.,  as  the  case  may  require,  and  point  off  the  quotient  as 
in  division  of  decimals.     (Art.  194.) 

9.  In  652254  square  meters,  how  many  hectares? 

10.  Reduce  87  meters  to  kilometers. 

11.  In  1482.35  grams,  how  many  kilograms? 

12.  In  39267.5  liters,  how  many  kiloliters? 

APPROXIMATE   VALUES. 

310  «•  111  comparing  Metric  Weights  and  Measures  with  those 
now  in  use,  the  approximate  values  are  often  convenient.  Thus, 
when  no  great  accuracy  is  required,  we  may  consider 

I  cu.  meter  or  stere  as     \  cord. 
I  liter  "       I  quart. 

I  hectoliter  "     2.^  bushels. 

I  gram  "    15^  grains. 

I  kilogram  "      2^  pounds, 

I  metric  ton  "  2200  pounds. 


I  decimeter  as        4  in 

I  meter 

5  meters 

I  kilometer 

I  sq.  meter 

I  hectare 


40  m. 

I  rod. 

f  mile. 

lof  sq.  feet. 

2  V  acres. 


309.  How  reduco  higher  metric  denomination*  to  lower  ?    310.  Lower  to  higher? 


214  METRIC      Oil     DECIMAL. 


APPLICATIONS    OF    METRIC    WEIGHTS      AND 
MEASURES. 

311.    To  add,  subtract,  multiply,  and  divide  Metric  Weights 
and  Measures. 
Rule. — Express  the   numbers   decimally,  and   inoceed 
as  in  the  corresponding  ojjerations  of  whole  numbers  and 
decimals. 

1.  What  is  the  sum  of  7358.356  meters,  8.614  hecto- 
meters, and  95  millimeters  ? 

Solution. — 7358.356  m. +  861.4111.,  +.095  m.=82i9.85i  m.  Ans. 

2.  What  is  the  differemce  between  8.5  kilograms  and 
976  grams? 

Solution. — 8.5  kilos— .976  ldlos=7.524  kilos.  Ans. 

3.  How  much  silk  is  there  in  12J  pieces,  each  contain- 
ing 48.75  meters? 

Solution. — 48.75  m.  x  12.5—609.375  m.  Ans. 

4.  How  many  cloaks,  each  containing  5.68  meters,  can 
be  made  from  426  meters  of  cloth  ? 

Solution. — 426  m.^5.68  in,  =  75  cloaks.  Ans. 

312.  To  reduce  Metric  to  Common  Weights  and  Measures. 

I.  Reduce  k.6  meters  to  inches. 

39.37  m. 

Analysis. — Since  in  i  meter  there  are  39.37  inches,  c  6  m 

in    5.6   meters   there  are   5.6   times  39.37  in. ;   and      ■ ~— 

39-37  X  56=220.472  in.     Therefore,  in  5.6  m.  there  ^?q    ^ 

are  220.472  in.     Hence,  the  ^9"85 

Ans.  220.472  in. 

Rule. — Multiply  the  value  of  the  principal  unit  of  ths 
Table  by  the  given  metric  number. 

Note. — Before  multiplying,  the  metric  number  should  be  reduced 
to  the  same  denomination  as  the  principal  unit,  whose  value  is 
taiLen  for  the  multiplicand. 

311.  How  are  Metric  Weights  and  Measures  added,  subtracted,  multiplied, 
and  divided  ?    312.  How  reduce  metric  to  common  weights  and  measures? 


WEIGHTS     A]SD     MEASURES.  215 

2.  In  45  Mlos,  how  many  pounds?  Ans.  99.207  lis. 

3.  In  6^  kilometers,  how  many  miles  ? 

4.  Reduce  75  liters  to  gallons. 

5.  Reduce  56  dekaliters  to  bushels. 

6.  Reduce  120  grams  to  ounces. 

7.  Reduce  137.75  kilos  to  pounds. 

8.  In  36  ares,  how  many  square  rods  ? 

A]srALYSis. — In  i  are  there  are  119.6  sq.  yds. ;  hence  in  36  ares  there 
are  36  times  as  many.  Now  119.6x36=4305.6  sq.  yds.,  and  4305.0 
sq.  yds, ^30^^=142.33  sq.  rods.  Ans. 

9.  In  60.25  hektars,  how  many  acres? 

10.  In  120  cu.  meters,  how  many  cu.  feet  ? 

313.  To  reduce  Common  to  Metric  Weights  and  Measures. 

11.  Reduce  213  feet  4  inches  to  meters. 
Analysis.— 213  ft.  4  in.= 2560  in.    Now,  opekatioh 

in  39.37  in.  there  is  i  meter;  therefore,  in      ^g.^7\2r6o!oo  in. 
2560  in.  there  are  as  many  meters  as  39.37  is  /  ~ — '- — 

contained  times  in  2560;  and  2560-7-39.37=  Ans,  65.02 -fm. 

65.02  +  metera    Hence,  the 

Rule. — Divide  the  given  numher  hy  the  value  of  the 
priiicijMl  inetric  unit  of  the  Table. 

Note. — Before  dividing,  the  given  number  should  be  reduced  t« 
the  lowest  denomination  it  contains ;  then  to  the  denomination  in 
which  the  vcdue  of  the  principal  unit  is  expressed. 

12.  In  63  yds.  3  qrs.,  how  many  meters? 

13.  Reduce  13750  pounds  to  kilograms. 

14.  Reduce  250  quarts  to  liters. 

15.  Reduce  2056  bu.  3  pecks  to  kiloiiters. 

16.  In  3  cwt.  15  lbs.  12  oz.,  how  many  kilos? 

17.  In  7176  sq.  yards,  how  many  sq.  meters? 

18.  In  40.471  acres,  how  many  hektars  ? 

19.  In  14506  cu.  feet,  how  many  cu.  meters? 

20.  In  36570  cu.  yards,  how  many  cu.  meters? 


313.  How  reduce  coramou  to  metric  weights  and  measures? 


0I»BRATION. 

lb.        OZ. 

pwt 

gr. 

13      7 

3 

18- 

9 

5 

6 

2      8^ 

8 

3 

compou:nd  addition. 

314.  To  add  two  or  more  Compound  Numbers. 

I.  What  is  the  sum  of  13  lb.  7  oz.  3  pwt.  18  gi\ ;  9  oz.  5 
pwt.  6  gr. ;  and  2  lb.  8  oz.  8  pwt.  3  gr.  ? 

Analysis. — Writing  the  same  denomi- 
nations one  under  another,  and  beginning 
at  the  right,  3  gr. +  6  gr.  +  i8  gr.  are  27 
gr.  The  next  higher  denomination  is 
pwt.,  and  since  24  gr.=i  pwt.,  27  gr. 
must  equal  I  pwt.  and  3  grains.  We  set  A7IS.  17  o  17  3 
the  3  gr.  under  the  column  added,  be- 
cause they  are  grains,  and  carry  the  i  pwt.  to  the  column  of  penny 
weights,  because  it  is  a  unit  of  that  column.  (Art.  30,  n.)  Next,  i 
pwt. +  8  pwt. +  5  pwt. +  3  pwt.  are  17  pwt.  As  17  pwt.  is  less  than 
an  ounce  or  a  U7iU  of  the  next  higher  denomination,  we  set  it  imder 
the  column  added.  Again,  8  oz.  +  9  oz.  +  7  oz.  are  24  oz.  Now  as 
12  oz.=i  lb.,  24  oz.  must  equal  2  lb.  and  o  oz.  We  set  the  o  under 
the  ounces,  and  carry  the  2  lb.  to  the  next  column.  Finally,  2  lb. 
+  2  lb.  + 13  lb.  are  17  lb.,  which  we  set  down  in  full.    Hence,  the 

Rule. — I.  Write  the  same  denominations  one  under 
another;  a7id,  beginning  at  the  rights  add  each  column 
separately. 

II.  If  the  sum  of  a  column  is  less  than  a  unit  of  the 
next  higher  denomination,  write  it  under  the  column  added. 

If  equal  to  one  or  more  units  of  the  next  higher  denomi- 
nation, carry  these  units  to  that  denominatioii,  and  write 
the  excess  under  the  column  added,  as  in  Simple  Additio7i. 

Notes. — i.  Addition,  Subtraction,  etc.,  of  Compound  Numbers  are 
the  same  in  principle  as  Simple  Numbers.  The  apparent  difference 
arises  from  their  scales  of  increase.  The  orders  of  the  latter  increase 
by  the  constant  scale  of  10  ;  the  denominations  of  the  former  by  a 
variable  scale.  In  both  we  carry  for  the  number  which  it  takes  of  a 
lower  ordier  or  denomination  to  make  a  unit  in  the  next  higher. 

2.  If  the  answer  contains  a  fraction  in  any  of  its  denominations, 
except  the  lowest,  it  should  be  reduced  to  wlwle  numbers  of  lower 
denominations,  and  be  added  to  those  of  the  same  name. 

314.  Ho^¥  are  oompound  numbers  added  ?  Ifote.  The  difference  between  Com- 
pound and  Simple  Addition  f    If  the  answer  contains  a  fraction  ? 


COM.'OtTKD     ADDITION.  217 

3  if  they  occur  in  the  given  numbers,  it  is  generally  best  to 
reduce  them  to  lowcjr  denominations  before  the  operation  is  com- 
menced. 

2.  What  is  the  sum  of  6  gal.  3  qt.  i  pt.  2  gi. ;  4  gal.  2  qt. 
o  pt  I  gi.,  and  7  gal.  i  qt.  i  pt.  3  gi.  ? 

Ans.  18  gaL  3  qt.  i  pt.  2  gi. 

(3-)  (4.)  (5-) 


£ 

s. 

d.    for. 

T. 

cwt. 

lb. 

oz. 

bu. 

pk. 

qt. 

7 

16 

6      I 

3 

6 

42 

4 

14 

2 

5 

I 

5 

8     2 

7 

0 

26 

7 

21 

3 

6 

2 

7 

9     3 

15 

17 

14 

8 

8 

I 

7 

6.  A  merchant  sold  19J  yd.  of  silk  to  one  lady;  16 J  yd. 
to  a  second,  20J  yd.  to  a  third,  and  lyf  to  a  fourth:  how 
much  silk  did  he  sell  to  all  ? 

7.  If  a  dairy  woman  makes  315  lb.  5  oz.  of  butter  in 
June,  275  lb.  10  oz.  in  July,  238  lb.  8  oz.  in  August,  and 
263  lb.  14  oz.  in  September,  bow  much  will  she  make  in 
4  months  ? 

8.  A  ship's  company  drew  from  a  cask  of  water  10  gals. 
2.qts.  I  pt.  one  day,  12  gal.  3  qt.  the  second,  15  gal.  i  qt. 
I  pt.  the  third,  and  16  gal.  3  qt.  the  fourth:  how  much 
water  was  drawn  from  the  cask  in  4  days  ? 

9.  A  farmer  sent  4  loads  of  wlieat  to  market;  the  ist 
contained  45  bu.  3  pk.  2  qt. ;  the  2d,  50  bu.  i  pk.  3  qt. ; 
the  3d,  48  bu.  2  pk.  5  qt. ;  and  the  4th,  51  bu.  3  pk.  5  qt. : 

_.  what  was  the  amount  sent  ? 

^^^  10.   How  much  wood   in    5  loads  which   contain   re- 
yT  spcctively  i  c.  28^  cu.  ft.;  i  c.  47 J  cu.  ft;  i  c.  ir  cu.  ft. 
I  c.  no  cu.  ft;  and  i  c.  iij  cu.  ft  ? 

II.  A  miller  bought  3  loads  of  whea,t,  containing  28  bu. 
3  pks. ;  ^6  bu.  i  pk.  7  qts. ;  S3  ^^-  2  pks.  3  qts.  respectively : 
how  many  bushels  did  he  buy  ? 

If  any  of  the  given  numbers  contain  a  fraction,  how  proceed?  315.  How  add 
denominate  fracticue  » 

10 


m. 

r. 

yd. 

ft. 

13 

65 

3 

I 

12 

40 

2 

I 

21 

19 

I 

2 

218  COMPOUND     ADDITIOl^. 

12.  What  is  the  sum  of  13  m.  65  r.  3  yd.  i  ft. ;  12  m. 
40  r.  2  yd.  I  ft. ;  and  21  m.  19  r.  i  3"d.  2  ft.? 

Solution. — In  dividing  tlie  yards 
"^75}  (tlis  number  required  to 
make  a  rod),  the  remainder  is  1^ 
yard.  Bat  i^  yd.  equals  i  ft.  6  in. ; 
we  therefore  add  i  ft.  6  in.  to  the  46    125      i-J   i 

feet  and  inches  in  the  result.     The  X  yd,   =z      01      6  in. 

Ans.  is  46  m.  125  r.  i  yd.  2  ft.  6 in.     ^^^g~(^    125      i      2      6 

13.  What  is  the  sum  of  8  w.  2^  d.  7  h.  40  m. ;  4  w.  54  d. 
o  h.  15  m. ;  and  10  w.  o  d.  16  h.  3  m.  ? 

14.  What  is  the  sum  of  15  A.  no  sq.  r.  30  sq.  yds.; 
6  A.  45  sq.  r.  16  sq.  yds  ;  42  A.  10  sq.  r.  25  sq.  yds.  ? 

315.  To  add  two  or  more  Denominate  Fractions. 

15.  What  is  the  sura  off  bu.  |  pk.  and  |  quart? 

Analysis -Reducing  the   de-        4  13^.^3  pk.   i  qt.   i  J  pt. 

nominate  fractions  to  whole  num-  ^  -j^-u-  .  ^ 

hers,    then    adding    them,    the        3q|-*__Q  q  ^i 

amount  is  3  pk.  6  qt.  g/tt  pt.,  Ans.  ;; 7 =— 

(Art.  292.)  ^  ^ " 

16.  What  is  the  sum  of  £.125,  .65s.  and  .75d.  ? 

Analysis. — Reducing  the  denominate  £.125  =2S.  6d.  ofal: 
decimals  to  whole  numbers    of   lower  .65s.  =0     7      3.2 

denominations,  then  adding,  the  result  .75d.  =  o     o      3 

is  3s.  2d.  2.2  far.     Hence,  the  j^^      7~~       ~ 

EuLE. — Rechice  the  denominate  fractions,  v)lietlier  coni' 
mon  or  decimal,  to  whole  numbers  of  lower  denominations: 
then  add  them  like  other  compound  numiers. 
17.  Add  .75  T.  .5  cwt.  .25  lb.     19.  Add  J  lb.  -^\  oz.  ^  pwt. 
:8.  Add  .25  bu.  ^  of  3  pk.        20.  Add  £.15,  .5s.  .8d. 

21.  Bought  4  town  lots,  containing  ^  acre;  |  acre; 
121^  sq.  rods;  and  150.5  sq.  rods  respectively:  how  much 
laud  was  there  in  the  4  lots? 

22.  Bought  3  loads  of  wood ;  one  containing  f  cord, 
another  75.3  cu.  feet;  the  other  ij  cord:  how  much  did 
they  all  contain  ? 


m. 

fur. 

r. 

ft. 

5 

3 

II 

4 

2 

I 

4 

9 

•  3 

2 

6 

Hi 

COMPOUNl)   SUBTEAOTIOK. 

316.  To  find  the  Difference  between  two  Compound  Numbers. 

1.  From  5  mi.  3  fur.  11  r.  4  ft.  take  2  mi.  i  fur.  4  r.  9  ft. 

Analysis. — We  write  the  less  number 
under  the  greater,  feet  under  feet,  etc. 
Beginning  at  the  right,  9  ft.  cannot  be 
taken  from  4  feet ;  we  therefore  borrow  i  Ans. 
rod,  a  unit  of  the  next  higher  denomina- 
tion. Now  I  rod,  or  i64  ft.,  added  to  4  ft.  are  20^  ft.,  and  9  ft.  from 
20 J  ft.  leave  11.^  ft.  Set  the  remainder  under  the  term  subtracted, 
and  carry  i  to  the  next  term  in  the  lower  number,  i  rod  and  4  r.  are 
5  r.,  and  5  r.  from  11  r.  leave  6  rods,  i  furlong  from  3  fur.  leaves 
2  fur:  ;  and  2  m.  from  5  m.  leave  3  m.     Therefore,  etc.     Hence,  the 

Rule. — I.  Write  the  less  number  under  the  greater, 
placing  the  same  denominations  one  under  another. 

II.  Beginnmg  at  the  right,  subtract  each  term  in  the 
loiver  number  from  that  above  it,  and  set  the  remainder 
under  the  term  subtracted. 

III.  If  any  term  in  the  lower  number  is  larger  than  that 
above  it,  borrow  a  unit  of  the  next  higher  denomination  and 
add  it  to  the  upper  term  ;  then  subtract,  and  carry  1  to  the 
next  term  in  the  lower  number,  as  in  Simple  Subtraction. 

Notes. — i.  Borrowing  in  Compound  Subtraction  is  based  on  the 
same  principle  as  in  Simple  numbers.  Tliat  is,  we  add  as 
many  units  to  the  term  in  the  upper  number  as  are  required  of 
that  denomination  to  make  a  unit  of  the  next  higher. 

2.  If  the  answer  contains  a  fraction  in  any  of  its  denominations 
except  the  lowest,  it  should  be  reduced  to  lower  denonfiinations,  and 
be  united  to  those  of  the  same  name,  as  in  Compound  Addition. 

2.  From  I  mile,  take  240  rods  3  yd.  2  ft. 

3.  From  14  lb.  5  oz.  3  pwt.  16  gr.,  take  7  lb.  6  pwt.  7  gr. 

4.  From  12  T.  9  cwt.  41  lb.,  take  5  T.  i  cwt.  15  lb. 

5.  Take  20  gal.  3  qt.  i  pt.  2  gi.  from  29  gal.  i  qt.  i  pt. 

316.  How  subtract  compound  numbers  ?    Note.  What  is  said  of  borrowing 
If  the  answer  contr-.ins  a  fraction,  how  proceed  ? 


220  COMPOUND     SUBTRACTION". 

6.  From  250  A-  45  sq.  r.  150  sq.  ft,  take  91  A.  32  sq.  r 
200  sq.  ft. 

7.  A  miller  has  two  rectangular  bins,  one  containing 
324  cu.  ft.  no  cu.  in.;  the  other,  277  cu.  ft.  149  cu.  in.: 
^hat  is  the  diff^ience  in  their  capacity  ? 

8.  A  man  having  320  A.  50  sq.  r.  of  land,  gave  his 
eldest  son  175  A.  29  sq.  r.,  and  the  balance  to  his  youngest 
son :  what  was  the  portion  of  the  younger  ? 

9.  A  merchant  bought  two  pieces  of  silk ;  the  longer 
contained  57  yd.  3  qr. ;  the  difference  between  them  was 
I  if  yards  :  what  was  the  length  of  the  shorter? 

10.  What  is  the  difference  between   10  m.  2  fur.  27  r. 


\  \     10.  vv  1 
\  I  ft.  7  in. 


and  6  m.  4  fur.  28  r.  i  yd.  ? 


317.  To  find  the   Difference   between   two   Denominate  Frac- 
tions. 

1.  From  I  of  a  yard,  take  ^J  of  a  foot. 

Analysis. — Reducing'  the  denominate  -|.  yd.  =  2  ft.  6  in. 

fractions  to   whole    numbers,   we    have  i-i  ft.    =  o  il  in. 

I  yard  =  2  ft.  6  in.  and  H  foot  =r    ii  in.  ~j^^           i  ft.  7  in. 
Ans.   I  ft.  7  in.     Hence,  the 

Rule. — Reduce  the  denominate  fractions,  whether  com- 
mon or  decimal,  to  luhole  7iumhers  of  lower  denominations; 
then  proceed  accordinrj  to  th<i  rule  above. 

2.  From  .£.525  take  .75  of  a  shilling. 

3.  From  J  bu.  take  f  of  a  peck. 

4.  From  J  of  a  gal.,  take  if  of  a  pint. 

5.  A  goldsmith  having  a  bar  of  gold  weighing  1.25  lbs., 
cut  off  sufficient  to  make  6  rings,  each  weighing  .15 
ounce ;  how  much  was  left  ? 

6.  If  a  man  has  .875  acre  of  land,  and  sells  off  two 
mn'llinT  lots,  each  containing  12.8  sq.  rods,  how  much 
land  will  be  left? 

7.  From  f  off  cwt.  of  sugar,  take  31.25  lbs 


317.  How  subtract  denominate  fractions  ? 


1870 

1776 

m. 
4 

7 

d. 
3 

4 

A71S.     93 

8 

2Q 

COMPOUI^D     SUBTRACTION^.  221 

318.  To   find   the    DilTepence   of  Time    between   two    Dates, 

in  years,  months,  and  days. 

I.  What  is  the  difference  of  time  between  July  4tli, 
1776,  and  April  3d,  1870  ? 

Analysis. — We  place  tlie  earlier  date  un- 
der the  later,  with  the  year  on  the  left,  the 
n  amber  of  the  month  next,  and  the  day  of 
the  month  on  the  right.  Since  we  cannot 
take  4  from  3,  we  borrow  30  days,  and  4  from 
33  leaves  29.  Carrying  i  to  the  next  denomination,  we  proceed  as 
above.     (Art.  264,  71.)     Hence,  the 

Rule. — Sei  tlw  earlier  date  under  the  later,  the  years  on 
the  left,  the  month  next,  and  the  day  on  the  right,  a7id  i^ro- 
ceed  as  in  suUracting  other  comjjoufid  numbers. 

Note. — Centuries  are  numbered  in  dates,  from  the  beginning  of 
the  Christian  era ;  months,  from  the  beginning  of  the  year ;  and 
days,  from  the  beginning  of  the  mouth. 

3.  Washington  was  born  Feb.  2 2d,  1732,  and  died  Dec. 
14th,  1799 :  at  what  age  did  he  die  ? 

319.  To  find   the   difference    between   two   Dates   In   Days, 

the  time  being  less  than  a  year. 

I.  A  note  dated  Oct.  20th,  1869,  was  paid  Feb.  loili. 

1870 :  how  many  days  did  it  run  ? 

Analysts.— In  Oct.  it  ran  31  —  20=^11 
days,  omittinp:  the  day  of  the  date;  in  Nov., 
30  days;  in  Dqq,.,  31  ;  in  Jan.,  31;  and  in 
Feb.,  10,  counting  the  day  it  was  paid. 
Now  II  +  30  +31  +  31  +  10=113  days,  the 
number  required.      Hence,  the 

Rule. — Set  doivn  the  numter  of  days  in  each  month  and 
part  of  a  month  lettveen  the  two  dates,  and  the  sum  ivill  he 
the  number  of  days  required. 

Note.  —The  day  on  which  a  note  or  draft  is  dated,  and  that  on 
which  it  becomes  due,  must  not  both  be  reckoned.  It  is  customary 
to  omit  i\iQ  former,  and  count  the  latter. 


Oct.  31- 

-20=11  d 

Nov. 

=  30  d. 

Dec. 

-31  d. 

Jan. 

=  31  d. 

Feb. 

=  10  d. 

Ans, 

113  d. 

31S.  How  find  the  difference  between  two  dates  in  years  ?    31  >  In  days ! 


222  COMPOUls^D     SUBTE  ACTION. 

2.  What  is  the  number  of  days  between  Xov.  loth,  1869, 
and  March  3d,  1870  ? 

3.  A  person  started  on  a  journey  Aug.  19th,  1869,  and 
returned  Nov.  ist,  1869:  how  long  was  he  absent? 

;    4.  A  note  dated  Jan.  31st,  1870,  was  paid  June  30th, 
1870:  how  many  days  did  it  run  ? 

5.  How  many  days  from  May  21st,  1868,  to  Dec.  31st, 
following  ? 

6.  Tiie  building  of  a  school-house  was  commenced  April 
I st,  and  completed  on  the  loth  of  July  following :  how 
long  was  it  in  building  ? 

320.  To  find  the  difference  of  Latitude  or  Longitude. 

Def.  t. — Latitude  is  distance  from  the  Equator.  It  is  reckoned 
in  degrees,  minutes,  etc.,  and  is  called  North  or  South  latitude,  ac- 
cording as  it  is  north  or  south  of  the  equator. 

2.  Longitiide  is  the  distance  on  the  equator  between  a  convene 
tional  or  fixed  meridian  and  the  meridian  of  a  given  place.  It  is 
reckoned  in  degrees,  minutes,  etc.,  and  is  called  East  and  West  longi- 
tude, according  as  the  place  is  east  or  icest  of  the  fixed  meridian,  lentil 
180^  or  half  the  circumference  of  the  earth  is  reached. 

7.  The  latitude  of  JSTew  York  is  40°  42'  43"  N. ;  that  of 
!New  Orleans,  29°  57'  30"  N.:  how  much  further  north  is 
New  York  than  New  Orleans  ?  * 

Analysis. — Placing  the  lower  latitude  [f^.  Y.  40°  42'  43'' 
under  the  higher,  and  subtracting  as  in  the  ^.  Q.  29°  57'  30" 
preceding  rule,  the  Ans.  is  10°  45'  13".  J^^g^    'io°'45'"i3^ 

8.  The  latitude  of  Cape  Horn  is  55°  59'  S.,  and  that  of 
Cape  Cod  42°  i'  57.1"  N. :  what  is  the  difference  of  lati- 
tude between  them  ? 

Analysis. — As  one  of  these  capes  is  north  of  the  equator  and  the 
other  south,  it  is  plain  that  the  difference  of  their  latitude  is  the  sum 
of  the  two  distances  from  the  equator.  We  therefore  add  the  two 
latitudes  together,  and  the  result  is  98°  o'  57  i'. 

320.  Z>«/".  What  19  latitude?  Longitude?  How  find  the  difference  of  latitude 
or  lono;itucle  between  two  })laces  ? 

*  The  latitude  and  longitude  of  the  places  in  the  United  States  here  given  ar« 
taken  from  the  American  Almanac,  1854;  those  of  places  in  foreign  lands,  mostly 
£ro;ii  Bo'^ditch's  Navigator. 


COMPOUND     SUBTRACTION.  223 

9.  The  longitude  of  Paris  is  2°  20'  East  from  Green- 
wich; that  of  Dublin  is  6"  20'  30"  West:  what  is  the  dif- 
ference in  their  longitude  ? 

Analysis. — As  tlie  longitude  of  one  of  tliese  places  2°  20'  00'' 

is  East  from  Greenwich  the  standard  meridian,  and  6°  20'  30" 

that  of  the  other  West,  the  difference  of  their  longi-  '^'aq'^-^q'' 

tude  must  be  the  sum  of  the  two  distances  from  the  '^ 
standard  meridian.     Ans.  8 '  40'  30".     Hence,  the 

Rule. — I.  If  hoth  places  are  the  same  side  of  the  equator, 
or  the  standard  meridian,  subtract  the  less  latitude  or 
longitude  from  the  greater. 

II.  If  the  places  are  on  different  sides  of  the  equator,  or 
the  standard  meridian,  add  the  two  latitudes  or  longitudes 
together,  and  the  sum  loill  he  the  ansicer. 

ro.  The  longitude  of  Berlin  is  13°  24'  E.  ;*  that  of 
Xew  Haven,  Ct.,  72°  55'  24''  W. :  what  is  the  difference  ? 

11.  The  longitude  of  Cambridge,  Mass.,  is  71°  7'  22"; 
that  of  Charlottesville,  Va.,  78°  31'  29":  what  is  the 
difference  ? 

12.  The  latitude  of  St.  Petersburg  is  59°  56'  north,  that 
of  Rome  41°  54'  north  :  what  is  their  difference  ? 

13.  The  latitude  of  Albany  is  4?°  2>9'  ^"y  ^^^t  of  Rich- 
mond 37°  32'  17":  what  is  the  difference? 

14.  The  latitude  of  St.  Augustine,  Flor.,  is  29°  48'  30"; 
that  of  St.  Paul,  Min.,  is'44°  52'  46":  what  is  the  dif- 
ference ? 

15.  The  longitude  of  Edinburgh  is  3°  12'  W. ;  that  of 
Vienna  16°  23'  E.f  :  required  their  difference  ? 

16.  The  latitude  of  Valparaiso  is  2>2>''  2'  S. ;  that  of  Ha- 
vana is  23°  9'  jST.:  what  is  the  difference? 

17.  The  latitude  of  Cape  of  Good  Hope  is  34°  22'  S. ; 
that  of  Gibraltar,  36°  7'  N. :  what  is  their  difference? 

18.  The  longitude  of  St.  Louis  is  90°  15'  16";  that  of 
Charleston,  S.  C,  79°  55'  38":  what  is  the  difference? 

*  Encyc.  Brit.  ■*■  Encyc.  Amer. 


COMPOUND   MULTIPLICATION. 

321.  To  multiply  Compound  Numbers. 

I.  A  farmer  raised  6  acres  of  wheat,  which  yielded  15  hn. 
3  pk.  I  qt.  per  acre:  how  much  wheat  had  he  ? 

Analysis. — 6  acres  will  produce  6  times  as  opekatioic. 

much  as  i  acre.     Beginning  at  tlie  right,  6  times  ^^-    P^-  ^*- 

1  qt.  are  6  qts.     As  6  quarts  are  less  than  a  ^'^      "^      a 

peck,  the  next  higher  denomination,  we  set  the  1 

6  under  the  term  multiplied.  6  times  3  pk.  are  A'^S.  94  2  6 
18  pk.     Since  4  pk.  =  i  bu.,  18  pecks=4  bu.  and 

2  pk.  over.  Setting  the  remainder  2  under  the  term  multiplied,  and 
carrying  the  4  bu.  to  the  next  product,  we  have  6  times  15  bu.-go 
bu.,  and  4  bu.  make  94  bu.     Therefore,  etc.     Hence,  the 

KuLE. — I.  Write  the  multiplier  under  the  lowest  denomi- 
nation of  the  multiplicand,  and,  leginning  at  the  rights 
multiply  each  term  in  succession. 

II.  If  the  product  of  any  term  is  less  than  a  itnit  of  the 
next  higher  denomination,  set  it  under  the  term  multiplied. 

III.  If  equul  to  one  or  more  units  of  the  next  higher  de- 
nomination, carry  these  units  to  that  denomination,  and 
write  the  excess  under  the  term  multiplied. 

Notes. — i.  If  the  multiplier  is  a  composite  number,  multiply  by 
one  of  the  factors,  then  this  partial  product  by  another,  and  so  on. 

2.  If  a  fraction  occurs  in  the  product  of  any  denomination  except 
the  lowest,  it  should  be  reduced  to  lower  deiiominations,  and  be 
united  to  those  of  the  same  name  as  in  Compound  Addition.   (Art.  3 1 4) 

(2.)  (3.) 

Mult.    12  T.  7  cwt.  16  lb.                 £21,  13s.  8Jd. 
By 8  7_ 

|.  What  is  the  weight  of  10  silver  spoons,  each  weigh- 
ing-3  oz.  7  pwt.  13  gr.? 

321.  How  are  compound  numbers  multiplied?  Nfce.  If  the  multiplier  is  a 
composite  number,  how  proceed  ?  When  a  fraction  occurs  In  ku;  denoninatiop 
except  the  last,  how  ? 


COMPOUND  MULTITLICATIOX.               226 

(5-)  (6-) 
Mult.  9  oz.  13  pwt.  7  gr.  by  i8.  Mult.  lo  r.  i  yd.  i  ft.  by  7. 

9  oz.  13  pwt.  7  gr.  10  r.  I  yd.  i  ft.  o  in. 

3  7 

2  lb.  4    19    21  71   3^   I 

6  Jycl.=        I   6 


14  lb.  5  oz.  19  pwt.  6  gr.  ^?^5.   ^?i5.  71  r.  3  yd.  2  ft.  6  in. 

7.  If  a  family  use  27  gal.  2  qt.  i  pt.  of  milk  in  a  montli, 
how  much  will  they  use  in  a  year  ? 

8.  If  a  man  chops  2  cords  67  cu.  feet  of  wood  per  day, 
how  much  will  he  chop  in  9  days  ? 

9.  What  cost  27  yards  of  silk,  at  17s.  7jd.  sterling  per 
yard? 

10.  If  a  railroad  train  goes  at  the  rate  of  23  m.  3  fur. 
2 1  r.  an  hour,  how  far  will  it  go  in  24  hours  ? 

11.  How  much  corn  will  6^  acres  of  land  produce,  at 
30  bu.  3  pk.  per  acre  ? 

12.  How  many  cords  of  wood  in  17  loads,  each  contain- 
ing I  cord  41  cu.  ft.  ? 

13.  How  much  hay  in  12  stacks  of  5  tons,  237  lbs.  each  ? 

14.  How  much  paper  is  required  to  print  20  editions  of 
a  book,  requiring  65  reams,  7  quires,  and  10  sheets  each  ? 

15.  If  a  meteor  moves  through  5°  2;^'  15''  in  a  second, 
how  far  will  it  move  in  30  seconds  ? 

16.  If  the  daily  session  of  a  school  is  5  h.  45  min.,  how 
many  school  hours  in  a  term  of  15  weeks  of  5  days  each  ? 

17.  A  man  has  11  village  lots,  each  containing  12  sq.  r. 
4  sq.  yd.  6  sq.  ft. :  how  much  do  all  contain  ? 

18.  If  I  load  of  coal  weighs  i  T.  48^  lb.,  what  will  72 
loads  weigh  ? 

19.  How  many  bushels  of  corn  in  12  bins,  each  con- 
taining 130  bu.  3  pk.  and  7  qt.  ? 

20.  A  grocer  bought  35  casks  of  molasses,  each  contain* 
ing  55  gal.  2  qt.  i  pt. :  how  much  did  they  all  contain? 


COMPOU]S"D   DIYISIOK 

322.  Division  of  Compound  Nuinbers,  like  Simple 
Division,  embraces  tico  classes  of  problems  : 

First. — Those  in  wliicli  the  dividend  is  a  compound  number,  and 
the  divisor  is  an  abstract  number. 

Second. — Those  in  which  both  the  divisor  and  dimdend  are  com- 
pound numbers.  In  the  former  the  quotient  is  a  compound  number^ 
In  the  latter,  it  is  times,  or  an  abstract  number.     (Art.  64.) 

323.  To  divide  one  Compound  Number  by  another,  or  by  an 
Abstract  number. 

Ex.  I.  A  dairy-woman  packed  94  lb.  2  oz.  of  butter  in 
6  equal  jars:  liow  much  did  each  jar  contain  ?• 

Analysis. — The  number  of  parts  is  given,  opeeation. 

to  find  the  value  of  each  part.     Since  6  jars  6)94  1^-  2  OZ. 

c:)ntain  94  lbs.  2  oz.,  i  jar  must  contain   \      Ans.     iq  lb.  11  OZ. 
of  94  lbs.  2  oz.     Now  \  of  94  lbs.  is  15  lbs. 

and  4  lbs.  remainder.     Reducing  the  remainder  to  oz.,  and  adding 
the  2  oz.,  we  have  66  oz.     Now  ^  of  66  oz.  is  11  oz.  (Art.  63,  b) 

2.  A  dairy-woman  packed  94  lbs.  2  oz.  of  butter  in  jars 
of  15  lbs.  II  oz.  each  :  how  many  jars  did  she  have? 

Analysts. — Here  the  size  of  each  94  lb.     2  oz.  =  1506  oz. 

part  is  given,  to  find  the  number  of  1 5  lb.  1 1  oz.  —     251  oz. 
parts  in  q4  lb.  2  oz.     Reduce  both  ^       OZ.)ic;o6  oz 

numbers  to  oz.,  and  divide  as  in  sitn-  ^  c.  •    ' 

pie  numbers.  (Art.  63,  a.)  Hence,  the  ^^^-  6  jars. 

fiuLE. — I.  When  the  divisor  is  an  abstract  number, 
Begiyming  at  the  left,  divide  each  de}iommation  in  suc- 
cessio7i,  and  set  the  quotient  under  the  term  divided. 

If  there  is  a  remainder,  reduce  it  to  the  next  lower  de- 
nomination, and,  adding  it  to  the  given  units  of  this  de- 
nomination, divide  as  before. 

11.  When  the  divisor  is  a  compound  number, 
Reduce  the  divisor  and  dividend  to  the  loivest  denomina- 
tion contained  in  either,  and  divide  as  in  simple  numbers. 


322.  How  many  classes  of  examples  does  division  of  compound  numbers  em 
bi-ace?    The  first?    The  second?    323.  What  i?  the  rule? 


TIME     AIN'D     LONGITUDE.  227 

Note. — If  the  divisor  is  a  composite  number,  we  may  diride  by  ite 
factors,  as  in  simple  numbers.     (Art.  77.) 

3.  Divide  29  fur.  19  r.  2  yd.  i  ft  by  7. 

4.  Divide  54  gal/  3  qt.  i  pt.  3  gi.  by  8. 

5.  A  miller  stored  450  bu.  3  pks.  of  grain  in  18  equal 
bins :  how  much  did  he  put  in  a  bin  ? 

6.  A  farm  of  360  A.  42  sq.  r.  is  divided  into  23  equal 
pastures:  how  much  land  does  each  contain? 

7.  How  many  spoons,  each  weighing  2  oz.  10  pwt.,  can 
be  made  out  of  5  lb.  6  oz.  of  silver  ? 

8.  How  many  iron  rails,  1 8  ft.  long,  are  required  for  a 
railroad  track  15  miles  in  length? 

9.  How  many  times  does  a  car-wheel  15  ft.  6  in.  in  cir- 
cumference turn  round  in  3  m.  25  r.  10  ft.  ? 

10.  How  many  books,  at  4s.  6Jd.  apiece,  can  you  buy  for 
£2,  14s.  3d.? 

11.  If  6  men  mow  ^6  A.  64  sq.  rods  in   6  days,  how 
much  will  I  man  mow  in  i  day  ? 

12.  A  farmer  gathered  150  bu.  3  pk.  of  apples  from  24 
trees :  what  was  the  average  per  tree  ? 


COMPARISON    OF   TIME   AND    LONGITUDE. 

324.  The  Earth  makes  a  revolution  on  its  axis  once  in  24 
hours ;  hence  ^V  part  of  its  circumference  must  pass  under  the  sun 
in  I  hour.  But  the  circumference  of  every  circle  is  divided  into 
360^,  and  -/4  of  360''  is  15°.  It  follows,  therefore,  that  15°  of  longi- 
tude make  a  difference  of  i  hour  in  time. 

Again,  since  15°  of  longitude  make  a  difference  of  i  hour  in  time, 
15'  of  longitude  {-^c^  of  15")  will  make  a  difference  of  i  minute  {-^^  of 
an  hour)  in  time. 

In  like  manner,  15"  of  longitude  (^Hr  of  15'),  will  make  a  difference 
of  I  second  in  time.     Hence,  the  following 

TABLE. 

15°  of  longitude  are  equivalent  to  i  hour  of  time. 
15                "             "            "  I  minute        " 

i.«5"  '*  "  "  I  second 


228  TIME     AXD     LONGITUDE. 

CASE    I. 

325.  To  find  the  Difference  of  Time  between  two^  places, 
the  difference  of  Longitude  being  given. 

Ex.  I.  Tlie  difference  of  longitude  between  'New  York 
and  London  is  73°  54'  3":  what  is  the  difference  of  time? 

Analysis. — Since  15°  of  Ion.  are  equiv-  Operation. 

alent  to  i  hour  of  time,  the  difference  of  15  )  73^  54   3'' 

time  must  be  -rs  part  as  many  hours,  min-      .  T    „  y- 

utes,  and  seconds  as  there  are  degrees,  -4    00      •  3   • 

etc.,  in  the  dif.  of  Ion. ;  and  73^  54'  3"-t-i5=:4  h.  55'  36.2"     Hence,  the 

Rule. — Divide  the  difference  of  longitude  by  15,  and  the 
degrees,  minutes,  and  seconds  of  the  quotient  will  le  the 
diff'erence  of  time  in  hours,  minutes,  and  secojids.    (Art.  323.) 

2.  The  difference  of  longitude  between  Savannah,  Ga., 
and  Portland,  Me.,  is  10°  53'  2":  what  is  the  difference  in 
time? 

3.  The  longitude  of  Boston  is  71°  3'  30'^  W.,  that  of 
Detroit  is  83°  2'  30"  W.:  when  it  is  noon  in  Boston  what 
is  the  time  at  Detroit  ? 

4.  The  longitude  of  Philadelphia  is  75°  9'  54",  that  of 
Cincinnati  84°  27':  when  it  is  noon  at  Cincinnati  what  is 
the  time  at  Philadelphia  ? 

5.  The  Ion.  of  Louisville,  Ky.,  is  8s°  30',  that  of  Bur- 
lington, Vt,  73°  10' :  what  is  the  difference  in  time  ? 

6.  When  it  is  noon  at  Washington,  wliat  is  the  time  of 
day  at  all  places  22°  30'  east  of  it?  What,  at  all  places 
22°  30'  west  of  it  ? 

7.  How  much  earlier  does  the  sun  rise  in  New  York, 
Ion.  74°  3",  than  at  Chicago,  Ion.  87°  35'? 

8.  How  much  later  does  the  sun  set  at  St.  Louis,  whose 
longitude  is  90°  15'  16"  W^,  than  at  Nashville,  Tenn., 
whose  longitude  is  86°  49'  3"  ? 

325.  How  find  the  diflFerence  of  time  between  two  places,  the  difference  ol 
longritnde  hein^f  i^iven  ? 


I 


tlME     AJfD     LONGITUDE.  ?>39 

CASE    II. 

326.   To  find  the  Difference  of  Longitude  between  two 
places,  the  difference  of  Time  being  given. 

9.  A  whaleman  wrecked  on  an  Island  in  the  Pacific, 
found  that  the  difference  of  time  between  the  Island  and 
San  Francisco  was  2  hr.  27  min.  54!  sec:  how  many  de- 
grees of  longitude  was  he  from  San  Francisco? 

Analysts. — Since  15°  of  Ion.  are  equira               Operatiok. 
lent  to  I  hour  of  time,  15'  of  Ion.  to  i  min.       2  h.  27  m.  54.6  seC. 
of  time,  and  15"  to  i  sec.  of  time,  there  must                               j  r 
be  15  times  as  many  de^rrees,  minutes,  and    

A  /COO''/ 

seconds  in  the  difference   of  longitude  as    -^^•^*  3"     5^    39 

there  are  hours,  minutes,  and  seconds  in  the 

difference  of  time  ;  and  2  h.  27  m.  54.6  s.  x  15=36°  58'  39".   Hence,  the 

'Rule.— Multiply  the  difference  of  time  hij  15,  ^nd  the 
hours,  minutes,  and  seconds  of  the  product  tvill  he  the 
difference  of  longitude  in  degrees,  minutes,  and  seconds. 

TO.  The  difference  of  time  between  Eichmond,  Ya.,  and 
JSTewport,  E.  I.,  is  24  min.  2>^  sec. :  what  is  the  difference 
of  longitude  ? 

1 1.  The  difference  of  time  between  Mobile  and  Galveston 
is  27  rain.  |  sec:  what  is  the  difference  of  longitude? 

12.  The  difference  of  time  between  Washington  and 
San  Francisco  is  3  hr.  i  min.  39  sec:  what  is  the  dif- 
ference in  longitude  ? 

13.  The  distance  from  Albany,  N.  Y.,  to  Milwaukee,  is 
nearly  625  miles,  and  a  degree  of  longitude  at  these  places 
is  about  44  miles :  how  much  faster  is  the  time  at  Albany 
than  at  Milwaukee  ? 

14.  The  distance  from  Trenton,  N.  J,  to  Columbus,  0., 
is  nearly  400  miles,  and  a  degree  of  longitude  at  these 
places  is  about  46  miles:  when  it  is  noon  at  Columbus 
what  is  the  time  at  Trenton  ? 

526.  How  find  the  difference  of  longitude,  when  the  difference  of  lime  's  glvwfl" 


PERCENTAGE. 

327  JPer  Cent  and  Unte  Per  Cent  denote  hun- 
dredths.  Thu^,  i  per  cent  of  a  number  is  yjo  part  of  that 
number ;  3  per  cent,  y^,  &c. 

328.  I^ercentage  U  the  resuU  obtained  by  finding  a 
certain  per  cent  of  a  number. 

Note. — The  term  per  cent,  is  from  the  Latin  per,  by  and  centum, 
hundred. 

NOTATION    OF    PER    CENT. 

329.  The  Sign  of  JPer  Cent  is  an  oUiqiie  line 
between  two  ciphers  {%) ;  as  3^,  15^. 

Note. — The  sign  (%),  is  a  modification  of  the  sign  division  (-r-), 
the  denominator  100  being  miderstood.     Thus  5  %  =-1^0  =  5-^100. 

330.  Since  per  cent  denotes  a  certain 7;ar^  of  a  hundred, 
it  may  obviously  be  expressed  either  by  a  common  fraction, 
wliose  denominator  is  100,  or  by  decimals,  as  seen  in  the 
following 

TABLE. 


3  per  cent  is  written  .03 

7  per  cent  "  .07 

10  per  cent  "  .10 

25  per  cent  "  .25 

50  per  cent  *'  .50 

100  per  cent  *'  i.oo 

125  per  cent  *'  1.25 

300  per  cent  "  3.00 


\  per  cent  is  written  .ocf^ 

\  per  cent  "  ,0025 

f  per  cent  "  -0075 

f  per  cent  "  .006 

2^  per  cent  "  .025 

7 1  per  cent  "  .074 

31^  per  cent  "  -3125 

112^  per  cent  "  1.125 


Notes. — i.  Since  hundredtJis  occupy  two  decimal  places,  it  follows 
that  every  per  cent  requires,  at  least,  two  decimal  figures.     Hence, 


327.  What  do  the  terms  per  cent  and  rate  per  cent  denote  ?  328.  What  is  per- 
centagre?  Note.  From  what,  are  the  terms  per  cent  and  percentage  derived? 
320.  What  is  the  sign  of  per  cent?  330.  How  may  per  cent  he  expressed? 
liote.  How  many  decimnl  places  does  it  require?  Why?  If  the  given  per  cent 
\8  less  than  10,  what  is  to  be  done  ?    Wliat  is  100  per  cent  of  a  numher  ? 


PERCEN-TAGE.  231 

\f  the  given  per  cent  is  less  than  lo,  a  cipher  must  be  prefixed  to 
the  figure  denoting  it.     Thus,  2  ^  is  written  .02  ;  6  %,  .06,  etc. 

2.  A  hundred  per  cent  of  a  number  is  equal  to  the  number  itself', 
for  |§iT  is  equal  to  i.     Hence,  100  per  cent  is  commonly  written  i.oo. 

If  the  given  per  cent  is  100  or  over,  it  may  be  expressed  by  an 
integer,  a  mixed  number,  or  an  improper  fraction.  Thus,  125  pei 
cent  is  written  125  %,  1.25,  or  |§f,.     Hence, 

331.  To  express  Per  Cent,  Decimally, 

Write  the  figures  denoting  tlie  per  cent  in  the  first  two 
places  on  the  right  of  the  decimal  point ;  and  those  denoting 
parts  of  I  jyer  cent,  in  the  succeeding  places  toward  the  right. 

Notes. — i.  When  a  given  part  of  i  per  cent  cannot  be  exactly 
expressed  by  one  or  two  decimal  figures,  it  is  generally  written  as  a 
common  fraction,  and  annexed  to  the  figures  expressing  the  integral 
per  cent.     Thus,  i\\fo  is  written  .04^,  instead  of  .043333  +  . 

2.  In  exi^ressing  per  cent,  when  the  decimal  point  is  used,  the 
words  per  cent  and  the  sign  (%)must  be  omitted,  and  mee  versa. 
Thus.  .05  denotes  5  per  cent,  and  is  eq-ual  to  too  or  2^1  \  t>ut  .05  per 
cent  or  .05  </c  denotes  tuu  of  too>  and  is  equal  to  tijuott  or  2inro. 

Express  the  following  per  cents, decimally: 

1.  2%,  6%,  Z%  14;^,  20^,  35^,  60^,  72^. 

2.  Zo%,  \o\%,  104%,  150^,  210^,  300^. 

3.  life,  4l%  H%,  H%,  loifc 

332.  To  read  any  given  Per  Cent,  expressed  Decimally. 

Read  the  first  two  decimal  figures  as  per  cent;  and  those 
on  the  light  as  decimal  parts  of  i  per  cent. 

Note. — Parts  of  i  per  cent,  when  easily  reduced  to  a  common 
fraction,  are  often  read  as  such.  Thus  .105  is  read  10  and  a  half  per 
cent ;  .0125  is  read  one  and  a  quarter  per  cent. 

Read  the  folio v/ing  as  rates  per  cent : 

4.  .05;  .07;  .09;  .045;  .0625;  .1875;  -125;  .165;  .27. 

5.  .10;  .17;  .0825;  .05125;  .ssl;  .i6|;  .75375- 

6.  i.co;  1.06;  2.50;  3.00;  1. 125;  1.0725;  1.83J. 


331.  How  express  per  cent,  decimally?  Note.  When  a  part  of  i  per  cent,  can- 
not be  exactly  expressed  by  one  or  two  decimal  figures,  how  is  it  commonly 
written  ?        332.  How  read  a  given  per  cent,  expressed  decimally  ? 


S'32  PERCENTAGE. 

333.  To   change   a   given    Per   Cent  from   a    Decimal   to   a 

Common   Fraction. 

7.  Change  5^  to  a  common  fraction.     Ans.  xSry^  A- 

8.  Change  .045  to  a  common  fraction.  A?is.  -^q. 

Rule. — Erase  the  decimal  point  or  sign  of  i^er  ceyit  (%), 
and  supply  the  required  denominator.     (Art.  179.) 

Note. — When  a  decimal  per  cent  is  reduced  to  a  common  fraction, 
then  to  its  lowest  terms,  this  fraction,  it  should  be  observed,  will 
express  an  equivalent  rate,  but  not  the  rate  per  cent. 

Change  the  following  per  cents  to  common  fractions : 

9.  5  percent;  10^;  4^;  207^;  25^;  50^;  75>t. 
10.  6i  per  cent;  12!^;  ^%^,  S3i%'y  62^^. 
/ \^  i  pel'  cent;  i%y  \%\  ifc,  i%]  Wo',  -hi\  25^. 

334.  To  change  a  common  Fraction  to  an  equivalent  Per  Cent. 

12.  To  what  per  cent  is  \  of  a  number  equal  ? 

Analysis. — Per  cent  dienoie^  hundredths.     The  J.  m  1.00-^3 

question  then  is, how  is  ^  reduced  to  hundredths?  i.oo—  -j -=  - •j ^ 
Annexing  ciphers  to  the  numerator,  and  dividing 

by  the  denominator,  we  have  ^  =  1.00-4-3  or  .33.^.  Hence,  the 

Rule. — An7iex  two  ciphers  to  the  numerator,  aiid  divide 
ly  the  denominator.     (Art.  186.) 

13.  To  what  ^  is  i  equal ?    J?    f?    i?    |?    |?    f? 

14.  To  what  %  is  ^\  equal?  tV?  ^V  ^V?  2V?  2V?  A? 

15.  To  whatsis  f  equal?    i?     i?    |?    i?    tV?    «? 

335.  In  calculations  of  Percentage,  four  elements  01 
'parts  are  to  be  considered,  viz. :  the  hase,  the  rate  per  cent, 
{he.  percentage,  and  the  amount. 

1.  The  base  is  the  number  on  which  the  percentage  is  calculated. 

2.  Tlie  ra^^  per  ccn^  is  the  number  which  shows  how  many 
hundredths  of  the  base  are  to  be  taken. 

331.  How  change  a  given  per  cent  from  a  decimal  to  a  common  fraction  ? 
334.  How  change  a  common  fraction  to  an  equal  per  cent  ?  335.  How  many  parlf 
&re  to  be  considered  in  calculptions  by  percenta;;e? 


PERCE  XT  AGE.  333 

3.  The  percentage  is  the  number  obtained  by  taking  that  portion 
of  the  base  indicated  by  the  rate  per  cent. 

4.  The  amount  is  the  base  plus,  or  minus  the  percentage. 

The  relation  between  these  parts  is  such,  that  if  any  two  of  them 
are  given,  the  other  two  may  be  found. 

Notes. — i.  The  term  amount,  it  will  be  observed,  is  here  employed 
in  a  modified  or  enlarged  sense,  as  in  algebra  and  other  departments 
of  mathematics.  This  avoids  the  necessity  of  an  extra  rule  to  meet 
the  cases  in  which  the  final  result  is  less  than  the  base. 

2.  The  conditions  of  the  question  show  whether  the  percentage  ia 
to  be  added  to,  or  subtracted  from  the  base  to  form  the  amount. 

3.  The  learner  should  be  careful  to  observe  the  distinction  between 
percentage  and  per  cent,  or  rate  per  cent. 

Percentage  is  properly  a  product,  of  which  the  given  per  cent  or 
rate  per  cent,  is  one  of  i'he  factors,  and  the  base  the  other.  This  cai^ 
is  the  more  necessary  as  these  terms  are  often  used  indiscriminately. 

4.  The  terms  |7er  cent,  rate  per  cent,  and  rate,  are  commonly  used 
as  synonymous,  unless  otherwise  mentioned. 

PROBLEM    I. 

336.  To   find   the    Percentage,  the    Base   and    Rate   being 

given. 

Ex.  I.  What  is  5  per  cent  of  $600  ? 

Analysis. — 5  per  ceift  is  .05  ;  therefore  5  per  $600  B. 

cent  of  a  number  is  the  same  as  .05  times  that  .05  K. 

number.   Multiplying  the  base,  $600,  by  the  rate     J[^g^  ~|^^o!oo  P. 
.05,  and  pointing  off  the  product  as  in  multipli- 
cation of  decimals,  the  result  is  $30.     (Art.  191.)    Hence,  the 

Rule. — Multiply  the  base  ly  the  rate,  exjjressed  deci- 
mally. 

Formula.    Percentage  —  Base  x  Pate. 

Notes. — i.  When  the  rate  is  an  aliquot  Y>SiTt  of  100,  the  percentage 
may  be  found  by  taking  a  like  part  of  the  base.  Thus,  for  20;^, 
take  i  ;  for  25%,  take  \,  etc.     (Arts.  105,  270.) 

2.  When  the  base  is  a  compound  number,  the  lower  denominations 
should  be  reduced  to  a  decimal  of  the  highest ;  or  the  higher  to  the 
lowest  denomination  mentioned  ;  then  apply  the  rule. 

3.  Finding  a  per  cent  of  a  number  is  the  same  as  finding  ^frac^ 
tionaZ  part  of  it,  etc.  The  pupil  is  recommended  to  review  with  care. 
Arts.  143,  165,  191. 

Explain  them.    What  is  the  relation  of  these  parts  ?    The  diflFerence  between  per- 
centage and  per  cent  ? 


234  PERCENTAGE. 

3.  s%  of  I807  ?  II.  sifc  of  1000  men  ? 

4.  5^  of  216  bushels?  12.  loj^  of  1428  meters? 

5.  8^  of  282.5  yds.  ?  13.  50$  of  $1715.57? 

6.  4^  of  216  oxen?  14.  ^^  of  £21.2  ? 

7.  5}^  of  150  yards  ?  15.  J;^  of  500  liters? 

8.  16%  of  $72.40?  16.  i%  of  230  kilograms? 

9.  1 2%  of  840  lbs.  ?  1 7.  100;^  of  840  pounds  ? 
10.  14^^  of  451  tons?  18.  200^  of  $500? 

19.  A  farmer  raised  875  bu.  of  corn,  and  sold  g^  of  it : 
how  many  bushels  did  he  sell  ?       7  ^  .  V  ^  4hv 

20.  The  gold  used  for  coinage  contains   10^^  of  alloy: 
how  much  alloy  is  there  in  3 1  pounds  of  standard  gold  ?  >  ^  ^ 

21.  A  man  having  a  hogshead  of  cider,  lost  15^5^  of  it 
by  leakage:  how  many  gallons  did  he  lose? 

^  22.  A  garrison  containing  4000  soldiers  lost  21^  of  them 
by  sickness  and  desertion  :  what  was  the  number  lost  ? 

23.  A  grocer  having  1925  pounds  of  sugar,  sold  12J  per 
cent  of  it :  how  many  pounds  did  he  sell  ? 

ANAiiYSls.— 12^  fo  is  i  of  1005,  and  100%  of  operation. 

a  number  is  equal  to  the  number  itself ;  there-  8)1925  Ibs. 

fore  iih  per  cent  of  a  number  is  equal  to  ^  of     ^^5      240.62c; 
that  number,  and  g^  of  1925  IbR.  is  240^  lbs.     In 
the  operation  we  take  ^  of  the  base. 

Solve  the  next  9  examples  by  aliquot  parts: 

24.  Find  25^  of  $86c.  26.   i2-|^  of  258  meters. 

25.  10;^  of  1572  pounds.         27.  20;^  of  580  liters. 

28.  A  drover  taking  2320  sheep  to  market,  lost  25^  of 
them  by  a  railroad  accident :  how  many  did  he  lose  ? 

29.  A  farmer  raised  468  bu.  of  corn,  and  ssi%  as  many 
oats  as  corn :  how  many  bushels  of  oats  did  he  raise  ? 

30.  A  young  man  having  a  salary  of  $1850  a  year,  spent 
50  per  cent  of  it:  what  were  his  annual  expenses? 

31.  What  is  33i-%  of  1728  cu.  feet  of  wood  ? 

32.  What  is  12  J  per  cent  of  £16,  8s.? 

•^^"S.  How  find  the  percentafre  when  the  ba!»e  and  rate  are  giv(»n  ?    When  tho 
rate  is  an  aliquot  part  of  loo,  how  proceed?    When  a  compound  number? 


PERCENT  AGE.  235 

PROBLEM    II. 
337.    To  find  the  Amount^  the  Base  and  Rate   being  given, 

1.  A  commenced  business  with  $1500  capital,  and  laid 
up  S%  the  first  year :  what  amount  was  he  then  worth  ? 

Analysis. — Since  he  laid  up   Sfc,  he  was  opekation. 

worth  his  capital,  $1500,  plus   8%    of  itself.  1 1 500    13. 

But  his  capital  is  100%  or  i  time  iteslf;  and.  i-o8?  i  H- R. 

100;^ +8;^  =108%   or  1.08;   therefore  he  was  120.00 

worth  1.08  times  1 1 500.     Now  $1500  x  1.08=  1500 

$1620.     We  multiply  the  base  by  i  plus  the  ^1620  00  Am't 
given  rate,  expressed  decimally.     (Art.  191.) 

2.  B  commenced  business  with  $1800  capital,  and 
squander  id  6%  the  first  year:  what  was  he  then  worth  ? 

Analysis. — As  B  squandered  6%,  he  was  $1800  B. 

worth  his  capital  $1800,    minus  6%   of  itself.  .q^  J^. 

But  his  capital  is  100^  or  i  time  itself;  and  .~ 

ioo%— 6%r=94%  or  .94.     Therefore  he  had  ,94  Tfloo 

times  $1800  :  and  $1800  x  .g4=:$i692.     Here  we  ^ .      , 

multiply  the  base  by  i  minus  the  given  rate,  $1692.00  Am  t 
expressed  decimally.     (Art.  191.)     Hence,  the 

Rule. — Multiply  the  base  by  1  plus  or  minus  the  rate,  as 
the  case  may  require.     The  result  will  be  the  amount. 

Formula.    Amount  =  Base  x  (i  ±  Bate), 

Note. — i.  The  character  (  ±  )  is  called  the  double  or  ambiguous 
8'gn.  Thus,  the  expression  $5  ±  $3  signifies  that  $3  is  to  be  added 
to  or  subtracted  from  $5,  as  the  case  may  require,  and  is  read, 
"  $5  plus  or  minus  $3." 

2.  The  rule  is  based  upon  the  axiom  that  the  whole  is  equal  to 
trie  sum  of  all  its  parts. 

3.  Wlien,  by  the  conditions  of  the  question,  the  amount  is  to  be 
greater  than  the  base,  the  multiplier  is  i  plus  the  rate  ;  when  the 
amount  is  to  be  less  than  the  base,  the  multiplier  is  i  minus  the 
rate. 

337.  How  find  the  amount  when  the  base  and  rate  are  given  ?  338.  How  else 
Ss  the  amount  fouad,  when  the  base  and  rate  are  given  ? 


UA 


236  PERCE:NrTAGE. 

338.  When  the  hase  and  rate  are  given,  the  amount  rirv 
also  be  obtained  by  fiv^tfindrngthepercentageytlien  adding 
it  to  or  subtracting  it  from  tJie  base.     (Art.  t,z^') 

3.  0  and  D  have  1000  sheep  apiece;  if  0  adds  i$%  to 
his  flock,  and  D  sells  12%  of  his,  how  manj^  sheep  will 
each  have  ? 

4.  A  merchant  having  I2 150.38  in  bank,  deposited 
']%  more :  what  amount  had  he  then  in  bank  ?       4  ^  % 

5.  If  you  have  $3000  in  railroad  stock,  and  sell  5^  of  it, 
what  amount  of  stock  will  you  then  have  ?  ^^  '  C.  5  ^ 

6.  The  cotton  crop  of  a  planter  last  year  was  450  bales; 
this  year  it  is  12  per  cent  more:  what  is  his  present  crop  ? 

7.  An  oil  well  producing  2375  gallons  a  day,  loses  15^ 
of  it  by  leakage :  what  amount  per  day  is  saved  ? 

8.  A  gardener  having  1640  melons  in  his  field,  lost  20^^ 
of  them  in  a  single  night:  what  number  did  he  have 
left  ? 

9.  A  man  paid  I420  for  his  horses,  and  12;^  more  for 
his  carriage :  what  was  the  amount  paid  for  the  carriage  ? 

10.  A  man  being  asked  how  many  geese  and  turkeys  he 
had,  replied  that  he  had  150  geese;  and  the  number  01 
turkeys  was  14^  less:  how  many  turkeys  had  he? 

11.  A  fruit  grower  having  sent  2500  baskets  of  peaches 
to  New  York,  found  g%  of  them  had  decayed,  and  sold  the 
balance  for  62  cts.  a  basket:  what  did  he  receive  for  hts 
peaches  ? 

12.  A  Floridian  having  4560  oranges,  bought  25^  mor-, 
and  sold  the  whole  at  4  cts*,  each :  what  did  he  receive 
for  them?  ^1^^ 

13.  If  a  man's  inc6tne  is  $7235  a  year,  and  he  spends 
ZZ\7o  of  it,  what  amount  will  he  lay  up  ?    • 

14.  A  man  bought  a  house  for  $8500,  and  sold  it  for 
2oi  more  than  he  gave :  what  did  he  receive  for  it  ?         -  -  '^ 

15.  A  merchant  bought  a  bill  of  goods  for  $10000,  and 
sold  them  at  a  loss  of  2l%\  what  did  he  receive  ? 


PERCENTAGE.  237 

PROBLEM     III. 

339.  To  find  the  Mate,  the  Base  and  Percentage  being  given; 

Or,  to  find  what  Per  Cent  one  number  is  of  another. 

I.  A  clerk's  salary,  being  1 1500  a  year,  was  raised  $25c', 
what  rate  was  the  increase  ? 

Analysis. — lu  this   example   $1500  is   the  operation. 

base,  and  $250  the  percentage.     The  question      1 500)^2^0^00  P. 
then  is  this:  $250  is  what  per  cant  of  $1500?  ^^5.   i6|-^^  E. 

iSTow  $250  is  iWo  of  $1500;  and  $250^11500 
=  .16666,  etc.,  or  i6|^.      The  first  two  decimal  figures  denote  the 
per  cent ;  the  others,  parts  of  i^.      (Arts,  331,  2.)     Hence,  the 

EuLE. — Divide  the  percentage  hy  the  base. 

Formula.  Bate  =  Percentage  ~  Base. 

Notes.— I.  This  prob.  is  the  same  as  finding  what  part  one  number 
is  of  another,  then  changing  the  common  fraction  to  hundredths. 
(Arts.  173,  186,  334-) 

It  is  based  upon  the  principle  that  percentage  is  a  product  o! 
which  the  base  is  a  factor,  and  that  dividing  a  product  by  one  of  its 
factors  will  give  the  other  factor.     (Art.  93.) 

2.  The  number  denoting  the  base  is  always  jireceded  by  the  word 
of,  which  distinguishes  it  from  the  percentage. 

3.  The  given  numbers  must  bo  reduced  to  the  same  denomination  ■, 
and  if  there  is  a  remainder  after  two  decimal  figures  are  obtained, 
place  it  over  the  divisor  and  annex  it  to  the  quotient. 

2.  What  ^  of  15  is  2  ?  6.  What  %  of  £3  are  1 5s.  ? 

3.  What  %  of  $20  are  $5  ?    7.  What  %  of  56  gals,  are  7  qts.? 

4.  What  %  of  48  is  16  ?        8.  What  %  are  5  dimes  of  $5  ? 

5.  W^hat  ^of  $5  are  75  cts.?  9.  What  ^  of  |  ton  is  I  ton  ? 
10.  The  standard  for  gold  and  silver  coin  in  the  IT.  S. 

is  9  parts  pure  metal  and  i  part  alloy :  what  %  is  the  alloy?         *| 
'  II.  From  a  hogshead  of  molasses  15  gals,  leaked  out;   f-^\-^t 
what  per  cent  was  the  leakage  ?  ^'^  J4 

12.  A  grocer  having  560  bbls.  of  flour,  sold  f  of  it. 
what  per  cent  of  his  flour  did  he  sell  ? 
_    13.  A  horse  and  buggy  are  worth  f>475 ;  the  buggy  is 
worth  $110;  what  %  is  that  of  the  value  of  the  horse  ? 

3^Q.  How  find  the  rate,  when  the  base  and  percentage  are  given  ?  To  what  id 
tills  problem  equivalent  ?    A'ow.  Upon  what  is  it  based  ? 


238 


P  E  Tv  C  E  is"  T  A  G  E  , 


PROBLEM    IV. 
340.  To  find  the  Base,  the  Percentage  and  Rate  being  given. 

1.  A  father  gave  his  son  $30  as  a  birthday  present, 
which  was  6%  of  the  sum  he  gave  his  daughter:  how 
much  did  he  give  his  daughter? 

Analysis. — The  percentage  $30  is  the  product  of       opekation. 
the  base  into  .06  the  rate;  therefore  $30-f-.o6  is  the      .o6)$30.oo 
other  factor  or  has3  ;  and  $30^. 06 =$500,  the  sum  he      Jiris7%<^oo 
gave  his  daughter.  (Art.  193,  n.) 

Or,  since  $30  is  6  %  of  a  number,  i  %  of  that  number  must  be  I  of 
$30,  which  is  $5  ;  and  100  fc  is  100  times  $5  or  $500. 

It. is  more  concise,  and  therefore  preferable,  to  divide  the  per- 
centage by  the  rate  expressed  decimally  ;  then  point  off  the  quotient 
as  in  division  of  decimals.     (Art.  193.)     Hence,  the 

EuLE. — Divide  the  percentage  hy  the  rate,  exp^^essed 
decimally. 

Formula.    Base  —  Percentage  -^  Bate. 

Notes. — i.  This  problem  is  the  same  as  finding  a  number  when 
a  given  per  cent  or  a  fractional  part  of  it  is  given.     (Arts.  174,  334.) 

2.  The  rule,  like  the  preceding,  is  based  upon  the  principle  that 
percentage  is  a  product,  and  the  rate  one  of  it^  factors.     (Art.  335,  n.) 

3.  Since  the  percentage  is  the  same  part  of  the  base  as  the  rate  is 
of  100,  when  the  rate  is  an  aliquot  part  of  100,  the  operation  will  \^ 
shortened  by  using  this  aliquot  part  as  the  divisor. 

2.  40  is  12^^  of  what  number  ? 
Solution. — i2y/c=^  and  40-7-^=40x8=320.  Ans. 

3.  20  =  5^  of  what  number?  Ans.  400. 

4.  15  bushels =6^  of  what  number  ? 


5.  $29  =  8;^  of  what? 

6.  45  tons  =  25^  of  what  ? 

7.  £150  =  33^^  of  what? 
^8.  3  7.5  =  6  Jf^  of  what? 

'9.  45  francs  =  1 2^%  of  what  ? 


10.  40=^^  of  what  ? 

11.  50  cts.=:-}^  of  what  ? 

12.  $100=1;^  of  what? 

13.  $35.20=1;:^  of  what? 

14.  68  yds.=  i25^of  what  ? 


340.  How  find  the  base,  when  the  percentage  and  rate  are  given  ?    Note.  Upon 
what  does  the  rule  depend  ?    When  the  rate  is  an  aliqnot  part  of  loo,  how  proceed  ? 


PEECENTAGE. 


23j 


15.  2;"^  of  $150  is  6^  of  Avliat  sum  ? 

16.  12^  of  500  is  60^  of  what  number? 

17.  A  paid  a  school  tax  of  $50,  which  was  i^  on  the 
valuation  of  his  property :  what  was  the  valuation  ? 

,  18.  B  saves  ^i^fo  of  his  income,  and  lays  up  $600:  what 
is  his  income  ? 

.  19.  A  general  lost  16}^  of  his  army,  315  killed,  no 
prisoners,  and  70  deserted;  how  many  men  had  he? 

20.  According  to  the  bills  of  mortality,  a  city  loses  450 
persons  a  month,  and  the  number  of  deaths  a  year  is  1^% 
of  its  population  :  what  is  its  population  ? 

PROBLEM    V. 
341.  To  find  the  Base,  the  Amount  and  Rate  being  given. 

1.  A  manufacturer  sold  a  carriage  for  $633,  which  wasJ 
5^^  more  than  it  cost  him :  what  was  the  cost  ? 

Analysis. — The  amount  received  $633,  is  operation. 

equal  to  the  cost  or  base  plus  5 [7%  of  itself.  1.055)1^633.000 
Now  the  cost  is  100^   or  i  time  itself,  and      Ans.  $600 

100% +5.J  %=i.o5^  ;  hence  $633  equals  1.05^ 

times  the  cost  of  the  carriage.  The  question  now  is;  633  is  105?^ 
or  1.05^  times  what  number?  If  633  is  1.05I  times  a  certain  num- 
ber, once  that  number  is  equal  to  as  many  units  as  i.osi  is  contained 
times  in  633  ;  and  633-7-1.055=1600.     Therefore  the  cost  was  $600. 

2.  A  lady  sold  her  piano  for  $628.25,  which  was  12-J^^, 
less  than  it  cost  her :  w  hat  was  the  cost  ? 

Analysis. — There  being  a  loss  in  this  case,  operation. 

the  amount  received,  $628.25,  equals  the  cost  .875)1)628.250 
or  base  wmw«  12^%  of  itself.  But  the  cost  is  ji^ns.  I718 
100%  or  I  time  itself,  and  100 f c— 12)1%— .^'jV,  ; 

hence  $628.25  equals  .87^  times  the  cost.  Now  if  $628.25  equals 
.87^  times  the  cost,  once  the  cost  must  be  as  many  dollars  as  .87.}  is 
contained  times  in  $628.25,  or  $718.     Hence,  the 

KuLE. — Divide  tJie  amount  hy  i  plus  or  miiius  the  rate,. 
as  the  case  may  require. 

Formula.     Base  —  Amount  -f-  (i  ±  Rate). 


341.  How  find  tlie  l.;ise  the  amount  and  rate  being  given  ?    Note.  Upon  wiiai 


24:0  PERCEiTTAGE. 

Notes. — i.  This  problem  is  the  same  as  finding  a  number  wliich 
is  a  gioen  per  cent  greater  or  less  than  a  given  number. 

2.  The  rule  depends  upon  the  principle  that  the  amount  is  a  vro- 
duct  of  which  the  base  is  one  of  the  factors,  and  i  plus  or  miuwi  the 
rate,  the  other. 

3.  The  nature  of  the  question  shows  whether  i  is  to  be  increased 
or  diminished  by  the  rate,  to  form  the  divisor. 

4.  When  the  rate  is  an  aliquot  part  of  100,  the  operation  is  often 
shortened  by  expressing  it  as  a  common  fT&ction.  Thus  25%=^; 
and  I  or  |  +  ^=f,  etc. 

3.  What  number  is  8^  of  itself  less  than  351?^.   325. 

4.  What  number  is  5 1^  of  itself  more  than  378  ?  A.  400. 

5.  What  number  diminished  33^^  of  itself  will  equal 

539-3-  ? 

6.  2275  is  25^  more  than  what  number? 

7.  I  is  12^^  more  than  what  number? 
Analysis.— i2i^=i ;  and  f -f-iir=f-j-|=f§  or  f.  Ans. 

8.  f  is  10^  less  than  what  number  ? 

9.  A  owns  I  of  a  ship,  which  is  i6f^  less  than  B's  part : 
what  part  does  B  own  ? 

10.  A  garrison  which  had  lost  28^  of  its  men,  had  3726 
left :  how  many  had  it  at  first  ? 

11.  A  merchant  drew  a  check  for  $4560,  which  was  25^ 
more  than  he  had  in  bank  :  how  much  had  he  on  deposit  ? 

12.  The  population  of  a  certain  place  is  8250,  which  is 
20^  more  than  it  was  5  years  ago :  how  much  was  it  then  ? 

13.  A  man  lays  up  $2010,  which  is  40^  less  than  his  in- 
'come :  what  is  his  income  ? 

14.  A  drover  lost  10^  of  his  sheep  by  disease,  1$%  were 
stolen,  and  he  had  171  left :  how  many  had  he  at  first  ? 

15.  The  attendance  of  9-  certain  school  is  370,  and  7^^ 
of  the  pupils  are  absent :  what  is  the  number  on  register  ? 

16.  An  array  having  lost  10^  in  battle,  now  contains 
5220  men  :  what  was  its  original  force  ? 

docs  this  rule  depend?  How  determine  whether  i  is  to  be  increased  or  dimin- 
ished by  the  rate  ?  When  the  rate  is  an  aliquot  part,  how  proceed?  To  what  19 
this  problem  equivalent  ? 


APPLIOATIOS'S    OF   PEECEI^TAGE. 

342.  The  JPrincij^les  of  Percentage  are  applied 
to  two  important  classes  of  problems: 

First.  Those  in  which  time  is  one  of  the  elements  0/ 
calculation ;  as,  Interest,  Discount,  etc. 

Second:  Those  which  are  independent  of  time ;  as.  Com- 
mission, Brokerage,  and  Profit  or  Loss. 

COMMISSION    AND    BROKERAGE. 

343.  Cormnission  is  an  allowance  made  to  agents, 
collectors,  brokers,  etc.,  for  the  transaction  of  business. 

I^roUer^age  is  Commission  paid  a  broker. 

Notes. — i.  An  Agent  is  one  who  transacts  business  for  another, 
and  is  often  called  a  Commission  Merchant,  Factor,  or  Correspondent. 

2.  A  Collector  is  one  who  collects  debts,  taxes,  duties,  etc. 

3.  A  Broker  is  one  who  buys  and  sells  gold,  stocks,  bills  of  ex- 
change, etc.  Brokers  are  commonly  designated  by  the  department 
of  business  in  which  they  are  engaged  ;  as,  Stock-brokers,  Exchange- 
brokers,  Note-brokers,  Merchandise-brokers,  Real-estate-brokers,  etc. 

4.  Goods  sent  to  an  agent  to  sell,  are  called  a  consignment ;  tho 
person  to  whom  they  are  sent,  the  consignee  ;  and  the  person  send- 
ing them  the  consignor. 

344.  Commission  and  Brolcerage  are  computed  at  a 
certain  per  cent  of  the  amount  of  business  transacted. 
Hence,  the  operations  are  precisely  the  same  as  those  in 
Percentage.     That  is. 

The  sales  of  an  agent,  the  sum  collected  or  invested  by 
him,  are  the  base. 
The  per  cent  for  services,  the  rate. 
The  commission,  the  percentage. 
The  sales,  etc.,  plus  or  minus  the  commission,  the  amount. 

342.  To  what  two  classes  of  problems  are  the  principles  of  percentage  applied? 
343.  What  is  commission?  Brokerage?  Note.  An  agent?  What  called?  A 
collector  ?  Broker  ?  344.  How  are  commission  and  brokerage  computed  ?  What 
Is  the  base?   The  rate?  The  percentage?   The  amount?    iVoie.  The  net  proceeds  ? 


242  COMMISSION     AND     BEOKERAGE. 

Notes. — i.  The  rate  of  commission  and  brokerage  varies.  Com* 
mission  merchants  usually  charge  about  2\  per  cent  for  seUi'n{j 
goods,  and  2i-  per  cent  additional  for  guaranteeing  the  payment, 
Btock-brckers  usually  charge  ^  per  cent  on  the  par  value  of  stocks, 
without  regard  to  their  market  value. 

2.  The  net  proceeds  of  a  business  transaction,  are  the  gross  amount 
of  sales,  etc.,  minus  the  commission  and  other  charges. 

345,    To  find   the   Conmiissionf  the  Sales    and   the   Rate 
being  given. 

Multiply  the  sales  ly  the  rate.     (Problem  I,  Percentage.) 

Notes. — i.  When  the  amount  of  sales,  etc.,  and  the  commission 
are  known,  the  net  proceeds  are  found  by  subtracting  the  commission, 
from  the  amount  of  sales.     Conversely, 

2.  When  the  net  proceeds  and  commission  are  known,  the  amount 
of  sales,  etc.,  is  found  by  adding  the  commission  to  the  7iet  proceeds. 

3.  When  both  the  amount  of  saies,  etc.,  and  the  net  proceeds  are 
known,  the  commission  is  found  by  subtracting  the  net  proceeds 
from  the  amount  of  sales. 

4.  In  the  examples  relating  to  stocks,  a  share  is  considered  $100, 
unless  otherwise  mentioned.     (Ex.  2.) 

(For  methods  of  analysis  and  of  dedvicing  the  rules  in  Commission, 
Profit  or  Loss,  etc.,  the  learner  is  referred  to  the  corresponding 
Problems  in  Percentage.) 

\j  I.  A  merchant  sold  a  consignment  of  cloths  for  $358: 
what  was  his  commission  at  2  J  per  cent  ? 
Solution. — Commission =$358  (sales)  x  .025  (rate)=$8.95.  Ans. 

2.  A  broker  sold  39  shares  of  bank  stock :  what  was  his 
brokerage,  at  J  per  cent  ? 

Solution. — 39  shares =$3900;  and  $3900  x  .005  =  $19,500.  Ans. 

3.  Sold  a  consignment  of  tobacco  for  $958.25  :  what  was 
ny  commission  at  2,i%  ? 

4.  A  man  collected  bills  amounting  to  $11268.45,  and 
sharged  2>i%'-  what  was  his  commission;  and  how  much 
did  he  pay  his  employer  ? 

345.  How  find  the  commleeion  or  brokerage,  when  the  sales  and  the  rate  arc 
ilwn  ?    iV»V,  How  find  the  net  nroceeds  ? 


COMMISSIOIT     Ai^D     BROKERAGE. 

5.  A  commission  merchant  sold  a  consignment  of  goods 
for  $4561,  and  cliarged  2^%  commission,  and  3^  for 
guaranteeing  the  payment:  what  were  the  net  proceeds? 

6.  An  agent  sold  1530  lbs.  of  maple  sugar  at  i6f  cts, 
for  which  he  received  2\%  commission:  what  were  the 
net  proceeds,  allowing  $7.50  for  freight,  and  $3,10  for 
storage  ? 

346.  To  find  the  Rate,  the  Sales  and  the  Commission 

being  given. 
Divide  the  commission  hy  the  sales.  (Prob.  III.  Percentage.) 

7.  An  auctioneer  sold  goods  amounting  to  $2240,  for 
which  he  charged  $53.20  commission :  what  per  cent 
was  that  ? 

Solution. — Per  cent  =  $53.20  (com.)  -^  $2240  (sales)  =  .02375,  or 
2%  per  cent.    (Art.  339,  331,  n.) 

8.  A  broker  charged  $19  for  selling  $3800  railroad 
stock:  what  per  cent  was  the  brokerage ? 

9.  Received  $350  for  selling  a  consignment  of  hops 
amounting  to  $7000:  what  per  cent  was  my  commis- 
sion ? 

10.  An  administrator  received  $118.05  ^^^  settling  an 
estate  of  $19675  :  what  per  cent  was  his  commission? 

347.  To  find   the  Sales,   the    Commission   and   the   Rate 

being  given. 

Divide  the  commission  ly  the  rate.     (Prob.  lY,  Per  ct.) 

11.  An  agent  charged  2%  for  selling  a  quantity  of 
muslins,  and  received  $93.50  commission :  what  was  the 
amount  of  his  sales  ? 

Solution.— Sales=:$93. 50  (com.)-^.o2  (rate)=$4675.  An8. 

346.  How  find  the  per  cent  commission,  when  the  sales  and  the  commission 
are  given  ?  347.  How  find  the  amount  of  sales,  when  the  commission  and  the 
rate  arc  ^von  ? 


244:  COMMISSION     AND     BROKERAGE 

12.  Received  $45  brokerage  for  selling  stocks,  which 
was  I-/0  of  what  was  sold:  what  was  the  amount  of  stocks 

S3ld? 

13.  A  commission  merchant  charging  2 1^  commission, 
and  2  ^/o  for  guaranteeing  the  payment,  received  $2 10.60  for 
selling  a  cargo  of  grain :  what  were  the  amount  of  sales, 
and  the  net  proceeds  ? 

14.  A  district  collector  received  $67.50  for  collecting  a 
school  tax,  which  was  4^%  commission :  how  much  did  he 
collect,  and  how  much  pay  the  treasurer  ? 

15.  An  auctioneer  received  $135  for  selling  a  house, 
which  was  1'^%:  for  what  did  tlie  house  sell;  and  how 
much  did  the  owner  receive  ? 

348.    To  find  the   Sales,  the   net   proceeds  and   per  cent 
commission  being  given. 

Divide  the  net  proceeds  hy  i  minus  the  rate.     (Prob.  V. ) 

16.  An  agent  sold  a  consignment  of  goods  at  2\%  com- 
mission, and  the  net  proceeds  remitted  the  owner  were 
$3381.30  :  what  was  the  amount  of  sales  ? 

Solution.— Sales=$338i. 30  (net  p.)-^-975  (i— rate) =$3468.  Ans. 

17.  A  tax  receiver  charged  5^  commission,  and  paid 
$4845  net  proceeds  into  the  town  treasury:  what  was  the 
am3unt  collected? 

Note. — In  this  and  similar  examples,  the  pupil  should  observe 
that  the  base  or  sam  on  which  commission  is  to  be  computed  is  the 
sum  collected,  aaJ  not  the  sunt  jjaid  ocer.  If  it  were  the  latter,  the 
agent  would  have  to  collect  his  own  commission,  at  his  own  expense, 
and  his  rate  of  commission  would  not  be  ytV,  but  j-^^.  In  the  cot 
lection  of  $100,000,  this  would  cause  an  error  of  more  than  $350, 

18.  After  retaining  2}%  for  selling  a  consignment  of 
flour,  my  agent  paid  me  $6664 :  required  the  amount  of 
yaljs,  and  his  commission. 

348.  How  find  the  sales,  etc.,  the  net  proceeds  and  the  per  cent  commission 
being  given  ? 


COMMISSIOif     A]S^D     BEOKERAGE.  245 

19.  After  deducting  ij^  for  brokerage,  and  $45.28  for 
advertising  a  house,  a  broker  sent  the  owner  $15250:  for 
what  did  the  house  sell  ? 

20.  An  administrator  of  an  estate  paid  the  heirs  $25686, 
charging  2^%  commission,  and  $350  for  other  expenses: 
what  was  the  gross  amount  collected  ? 

349.  To  find  the  Sum  Invesfedf  the  sum  remitted  and  the 
per  cent  commission  being  given. 

21.  A  manufacturer  sent  his  agent  $3502  to  invest  hi 
wool,  after  deducting  his  cemmission  of  3^ :  what  sum  dit  L 
lie  invest  ? 

AifALYSis. — The  sum  remitted  $3502,  includes  both  the  sum  in- 
vested and  the  commission.  But  the  sum  invested  is  100%  of  itself, 
and  100% +3%  (the  commission)  =103%.  The  question  now  is : 
$3502  is  103%  of  what  number?  $3502-7-1.03  — $3400,  the  sum  in- 
vested.    (Art.  340,  n.)     Hence,  the 

Rule. — Divide  the  sum  remitted  ly  1  plus  the  per  cent 
commission.     (Prob.  V,  Per  ct.) 

Note. — The  learner  will  observe  that  the  base  in  this  and  sim- 
ilar examples  is  the  sum  incested,  and  not  the  sum  remitted.  If  it 
were  the  latter,  the  agent  would  receive  commission  on  his  commis- 
sion, which  is  manifestly  unjust. 

„  22.  A  teacher  remitted  to  an  agent  $3131.18  to  be  laid 
out  in  philosophical  apparatus,  after  deducting  4%  com- 
mission :  how  much  did  the  agent  lay  out  in  appa- 
ratus ? 

23.  If  I  remit  my  agent  $2516  to  purchase  books,  after 
deducting  4%  commission,  how  much  does  he  lay  out  in 
books  ? 

24.  Remitted  $50000  to  a  broker  to  be  invested  in  city 
property,  after  deducting  1^%  for  his  services:  how  much 
did  he  invest,  and  what  was  his  commission  ? 


340.  How  find  the  sum  invested,  the  sum  remitted  and  the  per  cent  commis" 

sion  being  given? 


346 


ACCOUKT     or     SALES. 


ACCOUNT    OF    SALES. 

350.  An  Account  of  Sales  is  a  written  statement, 
made  by  a  commission  merchant  to  a  consignor,  contain- 
ing the  prices  of  the  goods  sold,  the  expenses,  and  the  net 
proceeds.     The  usual  form  is  the  following: 

Sales  of  Grain  on  accH  of  E.  D.  Barker,  Esq.,  Chicago. 


DATE. 

BUTEK. 

DESCRIPTION. 

BUSHELS. 

PRICE.  . 

EXTENSION. 

I87I. 

April   3 

J.  Hoyt, 

A.  Woodruff, 

Hecker  &  Co., 
Gross  am 

Winter  wheat. 
Spring       " 
Corn, 
:)unt. 

565 

870 

1610 

@  $2.10 
@       1.95 
@       1.05 

$1186.50 
1696.50 
1690.50 

$4573-50 

Charges. 


Freight  on  3045  bu.,  at  20  cts., 
Cartage  "  "       $15 -30, 

Storage  "         38-75. 


Commission, 


^ifo. 


$609.00 

1530 

38.75 

114.34 


Net  proceeds, 


_$777:39 
$3796.11 


New  York,  July  sth,  1S71 


J.  Henderson  &  Co. 


Bx.  25.  Make  out  an  Account  of  Sales  of  the  following: 

James  Penfield,  of  Philadelphia,  sold  on  account  of 
J.  Hamilton,  of  Cincinnati,  300  bbls.  of  pork  to  W.  Gerard 
&  Co.,  at  $27;  1150  hams,  at  I1.75,  to  J.  Eamsey;  875 
kegs  of  lard,  each  containing  50  lb.,  at  8  cts.,  to  Henry 
Parker,  and  750  lb.  of  cheese,  at  10  cts.,  to  Thomas  Young. 

Paid  freight,  $65.30;  cartage,  1 15.25  ;  insurance,  $6.45; 
commission,  at  2%.    \¥hat  were  the  net  proceeds  ? 

26.  Samuel  Barret,  of  New  Orleans,  sold  on  account  of 
James  Field,  of  St.  Louis,  85  bales  cotton,  at  I96.50; 
6s  barrels  of  sugar,  at  $48.25  ;  37  bis.  molasses,  at  $35. 

Paid  freight,  I45.50;  insurance,  I15  ;  storage,  $35.50; 
•jomraisRi^'m,  2^%.     What  were  the  net  proceeds? 


350.  What  is  an  accoant  of  Bales  ? 


PROFIT     AND     LOSS.  247 


PROFIT    AND    LOSS. 

351.  Profit  and  Loss  are  the  sums  gained  or  lost  in 
business  transactions.  They  are  computed  at  a  certain 
per  cent  of  the  cost  or  sum  invested,  and  the  operations 
are  the  same  as  those  in  Percentage  and  Commission. 

The  Cost  or  sum  invested  is  the  Base  ; 

The  Per  cent  profit  or  loss,  the  Rate  ; 

The  Profit  or  Loss,  the  Percentage  ; 

The  Selling  Price,  that  is,  the  cost  plus  or  minus  thu 
profit  or  loss,  the  Amount. 

352.  To  find  the  Profit  or  Loss,  the  Cost  and  the  Per  Cent 
Profit  or  Loss  being  given. 

Multiply  the  cost  hy  the  rate.     (Problem  1,  Per  ct.) 

Note.— When  the  'per  cent  is  an  aliquot  part  of  loo,  it  is  gener 
ally  shorter,  and  therefore  preferable  to  use  the  fraction.  (Art.  336,  n.) 

■  I.  A  man  paid  1 2 50  for  a  horse,  and  sold  it  at  15^ 
profit:  how  much  did  he  gain?  Ans.  $37.50. 

2.  A  man  paid  I450  for  a  building  lot,  and  sold  at  a 
loss  of  11^:  how  much  did  he  lose  ?  Ans.  I49.50. 

3.  Paid  $185  for  a  buggy,  and  sold  it  12^  less  than  cost: 
what  was  the  loss  ? 

4.  Paid  |i  10  for  a  pair  of  oxen,  and  sold  them  at  20^ 
advance :  what  was.  the  profit  ? 

5.  A  lad  gave  87^  cts.  for  a  knife,  and  sold  it  at  10% 
below  cost:  how  much  did  he  lose? 

6.  Bought  a  watch  for  I83I,  and  sold  it  at  a  loss  of  20^: 
what  was  the  loss  ? 

7.  Bought  a  pair  of  skates  for  I4.20,  and  sold  them  at 
Zd>i%  advance :  required  the  gain  ? 

351.  What  are  profit  and  loss?  How  reckoned?  To  what  does  the  cost  or 
sum  invested  answer?  The  per  cent  profit  or  loss?  The  profit  or  loss?  The 
selling  price  ?  352.  How  find  the  profit  or  loss,  when  the  cost  and  per  cent  ar« 
given  ?    Note.  When  the  per  cent  is  an  aliquot  part  of  100,  how  proceed  ? 


248  PROFIT     AKD     LOSS. 


353.    To  find  the  Selling  Frice,  the   Cost  and   Per  Cent 
Profit  or  Loss  being  given. 

Multiple/  the  cost  by  i  plus  or  minus  the  per  cent.  (Prob. 
II.  Percentage.) 

Note. — When  the  cost  and  per  cent  profit  or  loss  are  given,  the 
aeUing  price  may  also  be  found  by  first  finding  the  profit  or  loss; 
then  add  it  to  or  subtract  it  from  the  cost.     (Art.  338.) 

8.  A  man  paid  $300  for  a  house  lot :  for  what  must  he 
sell  it  to  gain  20^? 

Analysis. — To  gain  20%,  he  must  sell  it  for  the  cost  plus  20;^. 
That  is,  selling  pr.=$30o  (cost)  x  1.20  (i  +  20%) =$360.  Ans. 

9.  A  farmer  paid  I250  for  a  pair  of  oxen :  for  how 
much  must  he  sell  them  to  lose  15^? 

Selling  pr.=$25o  (cost)  x  .85  (i  — i5%)=$2i2.5o.  Ans. 

10.  A  and  B  commenced  business  with  $2500  apiece. 
A  adds  ij%  to  his  capital  during  the  first  six  months,  and 
B  loses  17^  of  his:  what  amount  is  each  then  worth  ? 

11.  A  merchant  paid  $378  for  a  lot  of  silks,  and  sold 
them  at  20^  profit:  what  did  he  get  for  the  goods ? 

12.  If  a  man  pays  $2750  for  a  house,  for  how  much 
must  he  sell  it  to  gain  y%  ? 

13.  If  a  man  starts  in  business  with  a  capital  of  I8000, 
and  makes  igfo  clear,  how  much  will  he  have  at  the  close 
of  the  year  ? 

14.  If  a  merchant  pays  15  cts.  a  yard  for  muslin,  how 
must  he  sell  it  to  lose  25^? 

15.  Bought  gloyes  at  $15  a  dozen:  how  must  I  sell 
them  a  pair,  to  lose  20^  ? 

16.  Paid  $25  per  dozen  for  pock'et  handkerchiefs:  for 
what  must  I  sell  them  apiece  to  make  ssi  per  cent? 

353.  How  find  the  Belling  price,  when  the  cost  and  per  cent  profit  or  loss  ara 
given.    Note.  How  else  may  the  selling  price  be  found  ? 


PROFIT     AND     LOSS.  249 

17.  Paid  $196  for  a  piece  of  silk  containing  50  yds.: 
how  must  I  sell  it  per  }'ard  to  gain  25^? 

18.  Bought  a  house  for  $3850:  how  must  I  sell  it  to 
make  i2|c^? 

19.  A  speculator  invested  $14000  in  flour,  and  sold  at  a 
loss  of  8-}%:  what  did  he  receive  for  his  flour? 

354.    To   find   the  Fer   Cent,  the   Cost  and   the  Profit  oi« 
Loss  being  given. 

Divide  the  profit  or  loss  hy  the  cost.     (Prob.  Ill,  Per  ct.) 

Note. — When  the  cost  and  selling  price  are  given,  first  find  the 
profit  or  loss,  then  the  2)er  cent.    (Art.  339.) 

20.  If  I  buy  an  acre  of  land  for  $320,  and  sell  it  for  $80 
more  than  it  cost  me,  what  is  the  per  cent  profit  ? 

Solution.— Per  cent=$8o  (gain)-H$32o  (cost)==.25  or  25%.  Arts. 

21.  A  jockey  paid  $875  for  a  fast  horse,  and  sold  it  so 
as  to  lose  $250 ;  what  per  cent  was  his  loss  ? 

22.  If  I  pay  22 J  cts.  a  pound  for  lard,  and  sell  it  at  2  J 
cts.  advance,  what  per  cent  is  the  profit  ? 

22,.  If  a  newsboy  pays  2^  cts.  for  papers,  and  sells  them 
at  I -J  cent  advance,  what  per  oput  is  his  profit? 

24.  If  a  speculator  buys  apples  at  $2.12}  a  barrel^  an(i 
sells  them  at  $2.87 J,  what  is  his  per  cent  profit? 

Analysis. — $2.87^  — $2. 12^=$. 75  profit  per  bl.     Therefore,  $.75 
(gain)-4-$2.i2i-  (cost)=.35,V  or  3Sri%-     (Art.  339,  n.) 

25.  If  I  sell  an  article  at  double  the  cost,  what  per  cent 
is  my  gain  ? 

26.  If  I  sell  an  article  at  half  the  cost,  what  per  cent  is 
my  loss  ? 

27.  If  I  buy  hats  at  $3,  and  sell  at  $5,  what  is  the  per 
cent  profit  ? 

28.  If  I  buy  hats  at  $5,  and  sell  at  $3,  what  is  the  per 
cent  loss  ? 

354.  How  find  the  per  cent  profit  or  loss,  when  the  cost  and  profit  or  lose  ara 
given  ? 


250  PROFIT     Al^D     LOSS. 

29.  If  a  man's  debts  are  I3560,  and  he  pays  only  $1780, 
what  per  cent  is  the  loss  of  his  creditors  ? 

30.  If  f  of  an  article  be  sold  for  ^  its  cost,  what  is  the 
per  cent  loss  ? 

Analysis. — If  f  are  sold  for  ^  its  cost,  ^  must  be  sold  for  ^  of  \, 
or  Jg  the  cost,  and  |  for  f  or  f  the  cost.  Hence,  the  loss  is  ^  the 
cost  -,  and  i-f-i=.333  or  sskfc 

31.  If  you  sell  ^  of  an  article  for  |  the  cost  of  the  whole, 
what  is  the  gain  per  cent  ? 

32.  If  I  sell  I  of  a  barrel  of  flour  for  the  cost  of  a  barrel, 
what  is  the  per  cent  profit  ? 

^:^.  If  a  milkman  sells  3  quarts  of  milk  for  tlie  price  he 
pays  for  a  gallon,  what  per  cent  does  he  make  ? 

34.  Bought  3  hhd.  of  molasses  at  85  cts.  per  gallon,  and 
sold  one  hhd.  at  75  cts.,  the  other  two  at  $1  a  gallon* 
required  the  whole  profit  and  the  per  cent  profit? 

355.  To  find  the  Cost^  the  Profit  or  Loss  and  the  Per  Cent 
Profit  or  Loss  being  given. 

Divide  the  profit  or  loss  hy  tlie  given  rate.  (Problem  IV, 
Percentage.) 

Note. — When  the  per  cent  profit  or  loss  is  an  aliquot  part  of 
100,  the  operation  may  often  be  abbreviated  by  using  this  aliquot 
part  as  the  divisor.    (Art.  341,  n) 

35.  A  grocer  sold  a  chest  of  tea  at  25^  profit,  by  which 
he  made  I22J:  what  was  the  cost  ? 

Solution.— Cost= $22.50 (profit)-T-. 25  (rate)=$90,  Ans. 
Or,  cost=$22.5o-r-^=$go.  Ans.     (Art,  340,  n.) 

36.  A  speculator  lost  $1950  on  a  lot  of  flour,  wiiich  was 
20;^  of  the  cost:  required  the  cost? 

37.  Lost  65  cents  a  yard  on  cloths,  which  was  13^  of 
the  cost:  required  the  cost  and  selling  price  ? 

Analysis.— The  cost=65cts.-^.i3=$5;  and  I5  — $.65=^4.35  the 
selling  price.     (Art.  340,  n.) 

355.  How  find  the  cost,  when  the  profit  or  loes  and  the  per  cent  profit  or  loss 

are  ijiven. 


PROFIT     AND     LOSS.  251 

38.  Gai  110(1  $2 J  per  barrel  on  a  cargo  of  flour,  which 
was  20;^:  requirod  the  cost  and  selling  price  per  barrel? 

Analysis. — 20%=:^^  and  $2.5o-i-^=$i2.5o  cost,  and  $12.50  + 
12.50=  $15,  selling  price. 

39.  If  I  sell  coffee  at  lo^  profit  I  make  lo  cts.  a  pound: 
what  was  the  cost  ? 

40.  A  man  sold  a  house  at  a  profit  of  33  J/^,  and  thereby 
gained  $7500:  required  the  cost,  and  selling  price? 

41.  If  I  make  20;^  profit  on  goods,  what  sum  must  I  lay 
out  to  clear  $3500,  and  what  will  my  sales  amount  to  ? 

42.  A  and  B  each  gained  $1500,  which  was  12^^  of  A*s 
and  16^  of  B's  stock :  what  was  the  investment  of  each  ? 

43.  If  a  merchant  sells  goods  at  10^  profit,  what  must 
be  the  amount  of  his  sales  to  clear  $25000  ? 

44.  A  market  man  makes  ^  a  cent  on  every  egg  he 
sells,  which  is  25^  profit:  what  do  they  cost  him,  and 
how  sell  them? 

356.  To  find   the  Cosf^  the   Selling  Price  and  the   Per  Cent 
Profit  or  Loss  being  given. 

Divide  the  selling  price  by  1  plus  or  minus  the  rate  of 
profit  or  loss,  as  the  case  may  require.     (Prob.  V,  Per  ct.) 

45.  A  jockey  sold  two  horses  for  $168.75  ^^^^^  ?  ^^  ^^^  ^^® 
made  i  zY/c,  on  the  other  lost  1 2\% :  what  did  each  horse 
cost  him  ? 

Solution.— Cost  of  one=$i68.75  (sel.  pr.)-f-i.i25  (i +rate)=$i50. 
Cost  of  other=  $168.75  (sel.  pr.)-4-.875  (i— rate)=$iq2.86.  Ans. 

46.  By  selling  568  bis.  of  beef  at  $15^  a  barrel,  a  grocer 
lost  12^%:  what  was  the  cost?  Ans.  $10061.714. 

47.  Sold  5000  acres  of  land  at  $3-J  an  acre,  and  thereby 
gained  22^:  what  was  the  cost? 

48.  Sold  a  case  of  linens  for  £27,  los.,  making  a  profit 
of  25^:  what  was  the  cost? 

356.  How  find  the  cost,  when  the  sslling  price  and  the  per  cent  profit  are  given  ? 


252  PROFIT     AND     LOSS. 

49.  If  a  newsboy  sells  papers  at  4  cts.  apiece,  lie  makes 
33i% '  what  do  they  cost  him  ? 

357.    To  Mark  Goods  so  that  a  given   Per  Cent  may  be 
deducted,  and  yet  make  a  given  Per  Cent  profit  or  loss. 

50.  Bought  shoes,  at  $2.55  a  pair:  at  what  price  must 
they  be  marked  that  15^  maybe  deducted,  and  jet  be 
sold  at  20^  profit  ? 

Analysis.— The  selling  price  is  120%  of  $2.55  (tlie  cost);  $2.55  x 
i.20=$3.o6,  the  price  at  which  they  are  to  be  sold.  But  the  marked 
price  is  100%  of  itself;  and  ioo%— 15%=85%.  The  question  now 
IS,  $3.06  are  85^  of  what  sum  V  If  11^3.06 =-i^y%,  7^7;  =  $3.06 ^85,  or 
^.036  ;  and  |{}{l=$3.6o.  Ans.    (Art.  340.)    Hence,  the 

EuLE. — Find  the  selling  price,  and  divide  it  dy  1  minus 
the  given  per  cent  to  he  deducted ;  the  quotient  will  he  the 
marhed  price, 

51.  Paid  56  cts.  apiece  for  arithmetics:  what  must  they 
be  marked  in  order  to  abate  5;^,  and  yet  make  25%? 

52.  If  mantillas  cost  $24  apiece,  at  what  price  must  they 
be  marked  that,  deducting  8^,  the  merchant  may  realize 
33j^  profit? 

53.  When  apples  cost  I3.60  a  barrel,  what  must  be  the 
asking  price  that,  if  an  abatement  of  1 2^%  is  made,  there 
will  still  be  a  profit  of  i6|^^? 

54.  A  merchant  paid  87^  cts.  a  yard  for  a  case  of  linen, 
which  proved  to  be  slightly  damaged :  how  must  he  mark 
it  that  lie  may  fall  25^^,  and  yet  sell  at  cost  ? 

55.  A  goldsmith  bought  a  case  of  watches  at  $60:  how 
must  he  mark  them  that,  abating  ^%,  he  may  make  20^? 

56.  If  cloths  cost  a  tailor  $4.50  a  yard,  at  what  price 
must  he  mark  them,  that  deducting  10^  he  will  make 
15  per  cent  profit? 

357.  When  the  selling  price  and  the  per  cent  profit  or  loss  are  given,  how  find 
the  per  cent  profit  or  loss  .nt  any  proposed  price  ?  How  find  what  to  mark  goods, 
that  a  given  per  cent  m.xy  be  deducte  J,  and  yet  a  given  per  cent  profit  or  loss  be 
-.flAli^ed  "i 


PKOriT    ANL     LOSS.  253 

QUESTIONS    FOR    REVIEW. 

1.  A  grocer  paid  23  cts.  a  lb.  for  a  tub  of  butter  weighing 
65  lbs.,  and  sold  it  at  18^  profit:  hoAV  much  did  he  make? 

2.  Bought  a  house  for  I3865,  and  paid  $1583.62  for  re- 
pairs: how  mucli  must  I  sell  it  for  to  make  i6f^? 

3.  Paid  $2847  for  ^  case  of  shawls,  and  $956  for  a  case 
of  ginghams;  sold  the  former  at  22^^  advance,  and  the 
latter  at  11^  loss :  what  was  received  for  both  ? 

4.  A  jockey  bought  a  horse  for  $125,  which  he  traded 
for  another,  receiving  $37  to  boot;  he  sold  the  latter  at 
25^  less  than  it  cost:  what  sum  did  he  lose  ? 

5.  What  will  750  shares  of  bank  stock  cost,  if  the  bro- 
kerage is  1%'  and  .the  stock  2>%  above  par  ? 

6.  Sold  3  tons  of  iron,  at  15 1  cts.  a  pound,  and  charged 
2,1%  commission  :  what  were  the  net  j)roceeds  ? 

7.  Bought  wood  at  $4^  a  cord,  and  sold  it  for  %G\:  re- 
quired the  per  cent  profit  ? 

8.  A  shopkeeper  buys  thread  at  4  cts.  a  spool,  and  sells 
at  6 J  cts.:  what  per  cent  is  his  profit,  and  ho\?  much 
would  he  make  on  1 000  gross  ? 

9.  Bought  1650  tons  of  ice,  at  $12  ;  one  half  molted,  and 
sold  the  rest  at  $1  a  hund. :  what  per  cent  was  the  loss? 

10.  .The  gross  proceeds  of  a  consignment  of  apples  was 
$1863.75,  and  the  agent  deducted  $96,911  for  selling:  re- 
quired the  per  cent  commission  and  the  net  jiroceeds? 

11.  A  lady  sold  her  piano  for  J  of  its  cost:  what  per 
cent  was  the  loss  ? 

12.  If  I  pay  $2  for  3  lbs.  of  tea,  and  sell  2  lbs.  for  $3, 
what  is  the  per  cent  profit  ? 

13.  A  man  sold  his  house  at  20^  above  cost,  and  thereby 
made  $1860:    required  the  cost  and  the  selling  price? 

14.  A  miller  sells  flour  at  15;^  more  than  cost,  and 
makes  $1.05  a  bar.:  what  is  the  cost  and  selling  price  ? 

15.  Lost  25  cts.  a  pound  on  indigo,  which  was  12^%  of 
the  cost :  required  the  cost  and  the  selling  price. 


254  PROFIT     AN^t)     LOSS. 

1 6.  A  commission  merchant  received  $260  for  selling 
a  quantity  of  provisions,  which  was  5^:  required  the 
amount  of  sales  and  the  net  proceeds. 

17.  A  broker  who  charges  i%,  received  |6o  for  selling  a 
quantity  of  uncurrent  money  :  how  much  did  he  sell  ? 

18.  A  grocer  makes  $1.25  a  pound  on  nutmegs,  which 
is  100^  profit :  what  does  he  pay  for  them  ? 

19.  A  merchant  sold  a  bill  of  white  goods  for  $7000, 
and  made  ssl% :  required  the  cost  and  the  sum  gained. 

20.  Sold  a  hogshead  of  oil  at  93J  cts.  a  gallon,  and 
made  i8J^:  required  the  cost  and  the  profit. 

21.  A  man,  by  selling  flour  at  $i2|  a  barrel,  makes  25^: 
what  does  the  flour  cost  him  ? 

22.  Made  12^^  on  dry  goods,  and  the  amount  of  sales 
was  $57725  :  required  the  cost  and  the  sum  made. 

23.  Sold  a  quantity  of  metals,  and  retaining  3j^  com- 
mission, sent  the  consignor  $15246:  required  the  amount 
of  the  sale  and  the  commission. 

24.  Sold  250  tons  of  coal  at  $6-|,  and  made  12^^:  what 
per  cent  would  liave  been  my  profit  had  I  sold  it  for 
$8  a  ton  ? 

25.  A  merchant  sold  out  for  $18560,  and  made  15;^  on 
his  goods :  what  per  cent  would  he  have  gained  or  lost  by 
selling  for  $15225  ? 

26.  Paid  $40  apiece  for  stoves :  what  must  I  ask  that  I 
may  take  off  20^,  and  yet  make  20;^  on  the  cost  ? 

27.  If  a  bookseller  marks  his  goods  at  25^  above  cost, 
and  then  abates  25^,  what  per  cent  does  he  make  or  lose  ? 

28.  A  grocer  sells  sugar  at  2|  cts.  a  pound  more  than 
cost,  and  makes  20%  profit :  required  the  selling  price. 

29.  A  man  bought  2500  bu.  wheat,  at  $if ;  3200  bu.  corn, 
at  87  J  cts. ;  4000  bu.  oats,  at  25  cts.;  and  paid  $450  freight: 
he  sold  the  wheat  at  5;^  profit,  the  corn  at  11%  loss,  and 
the  oats  at  cost ;  the  commission  on  sales  was  $%  •  what 
was  his  per  cent  profit  or  loss  ? 


INTEREST. 

358.  Interest  is  a  compensation  for  tlie  use  of  money. 
It  is  computed  at  a  certain  per  cent,  per  annum,  and  em- 
braces five  elements  or  parts :  tlie  Principal,  tlie  Bate,  the 
Interest,  the  Time,  and  the  Amount. 

359.  The  I^rincij^al  is  the  money  lent. 
The  Mate  is  the  per  cent  per  amimn. 
The  Interest  is  the  percentage. 

The  Time  is  the  period  for  which  the  principal  draws 
interest. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

Notes. — i.  The  term  per  annum,  from  the  Latin  per  and  annus, 
signifies  b^/  the  year. 

2.  Interest  differs  from  the  preceding  applications  of  Percentage 
only  by  introducing  time  as  an  element  in  connection  with  per  cent. 
The  terms  rate  and  rate  per  cent  always  mean  a  certain  number  of 
hundredths  yearly,  and  pro  rata  for  longer  or  shorter  periods. 

360.  Interest  is  distinguished  as  Simple  and  Compound. 
Simple  Interest   is   that  which   arises   from  the 

principal  only. 

Compound  Interest  is  that  which  arises  both  from 
the  principal  and  the  interest  itself,  after  it  becomes  due. 

361.  The  Legal  Mate  of  interest  is  the  rate  allowed 
by  law.    Rates  higher  than  the  legal  rate,  are  called  usury. 

Notes. — i.  In  Louisiana  the  legal  rate  is     -        -  -      5%- 

In  the  N.  E.  States,  N.  C,  Penn.,  Del.,  Md.,  Va.,  W.  Va., 
Tenn.,  Ky.,  0.,  Mo.,  Miss.,  Ark.,  Flor.,  la.,  111.,  Ind.,  the 
Dist.  of  Columbia,  and  debts  due  the  United  StatttS,     -      ^%. 
In  N.  Y,  N.  J.,  S.  C,  Ga.,  Mich.,  Min.,  and  Wis.,        -  -]%. 

In  Alabama  and  Texas,  -        -        -        -         -        -        -      8%, 

In  Col.,  Kan.,  Neb.,  Nov.,  Or.,  Cal.  and  Washington  Ter.,  -     lofo. 

2.  In  some  States  the  law  allows  higher  rates  by  special  agreement. 

3.  When  no  rate  is  specified,  it  is  understood  to  be  the  legal  rate. 


358.  What  is  interest?    359.  The  principal ?    The  rate?    The  int.  ?    The  time? 
The  amt.  ?    360.  What  is  simple  interest  ?    Compound  ? 


356  INTEREST. 

THE    SIX    PER    CENT    METHOD. 

362.  Since  the  interest  of  $i  at  6  per  cent  for  1 2  months 
or  I  year,  is  6  cents,  for  i  month  it  is  i-hoelfth  of  6  cents 
or  ^  cent ;  for  2  months  it  is  2  halves  or  i  cent ;  for  3  mos. 
I J  cent,  for  4  mos.  2  cents,  etc.     That  is, 

The  interest  of  $1,  at  6  per  cent,  for  any  number  of 
mo7iths,  is  half  as  many  cents  as  months. 

363.  Since  the  interest  of  $1  at  6  per  cent  for  30  days 
or  I  mo.,  is  5  mills  or  |  cent,  for  i  day  it  is  -j-q  of  5  mills 
or  J  mill ;  for  6  days  it  is  6  times  J  or  i  mill ;  for  1 2  days, 
2  mills ;  for  15  days,  2^  mills,  etc.     That  is. 

The  interest  of  ^1,  at  6  per  cent,  for  any  number  of  days. 
is  i-sixth  as  many  mills  as  days.     Hence, 

364.  To  find  the  Interest  of  $1,  at  6  per  cent,  for  any 
given  time. 

Take  half  the  number  of  months  for  cents,  and  one  sixth  of 
the  days  for  mills.     Their  sum  will  be  the  interest  required. 

Notes. — i.  When  the  rate  is  greater  or  less  than  6  per  cent,  the 
interest  of  $1  for  the  time  is  equal  to  the  interest  at  6  per  cent 
increased  or  diminished  by  such  a  part  of  itself  as  the  given  rate 
exceedjS  or  falls  short  of  6  per  cent.  Thus,  if  the  rate  is  ']%,  add  \  to 
the  int.  at  6;/ ;  if  the  rate  is  5^',  subtract  I,  etc. 

2.  In  finding  i-sixth  of  the  days,  it  is  sufficient  for  ordinary  pur- 
poses to  carry  the  quotient  to  tenths  or  hundredths  of  a  mill. 

3.  When  entire  accuracy  is  required,  the  remainder  should  be 
placed  over  the  6,  and  annexed  to  the  multiplier. 

1.  Int.  of  81,  at  "]%  for  9  m.  18  d.  ?  A71S.  $.056. 

2.  Int.  of  $1,  at  s%  for  11  m.  21  d.  ?       Ans.  I.04875. 

3.  What  is  the  int.  of  Si,  at  6%  for  7  m.  3  d.  ? 

4.  What  is  the  int.  of  $1,  at  6%  for  11  m.  13  d.  ? 

Note.  Wherein  does  intereet  differ  from  the  preceding  applications  of  per- 
centage ?  361.  The  legal  rate  ?  What  is  ueuiy  ?  362.  What  is  the  interest  of  $1 
nt  6  per  cent  for  months  ?  363.  For  days  ?  364.  IIow  find  the  int.  of  $1  for  nny 
[riven  time  ?  Note.  When  the  rate  is  greater  or  lees  than  6  per  cent,  to  what  id 
the  intereet  of  f  i  equal? 


IKTEKEST.  267 

Rem. — The  relation  between  the  principal,  the  interest,  the  rate, 
the  time,  and  the  amount,  is  such,  that  when  any  three  of  them  are 
given,  the  others  can  "be  found.  The  most  important  of  these 
problems  are  the  following : 

PROBLEM    I. 

365.    To  find  the  Interest,  the   Princ  pal,  the   Rate,  and 
Time  being  given. 

1.  What  is  the  interest  of  $150.25  for  i  year  3  months 
and  18  days,  at  6^? 

Analysis. — The  interest  of  $1  for  15  m.=.o75  operation. 

18  d.  =.003  Prin.  $150.25 

I  7.  3  m.  18  d.  =^78"  -oyS 

Now  as  the  int.  of  $1  for  the  given  time  and  120200 

rate  is  $.078  or  .078  times  the  principal,  the  int.  105  175 

of  $150.25   must   be  .078  times   that   sum ;  and  %ii.niq^ 
$150.25  X  .o78  =  $ii. 71950.     Hence,  the 

EuLE. — Multiply  the  ^^rmc'?};^^/  ly  the  interest  of  $1  for 
the  time,  expressed  decimally.     (Art.  336.) 

For  the  amount,  add  the  interest  to  the  priyicijxd. 

Rem. — I.  The  amount  may  also  be  found  by  multiplying  tha 
principal  by  i  plus  the  interest  of  %i  for  the  time.     (Art,  337.) 

2.  When  the  rate  is  greater  or  less  than  6%,  it  is  generally  best  to 
find  the  interest  of  the  principal  ai  6%  for  the  given  time  ;  tlien  add 
to  or  subtract  from  it  such  a  part  of  itself,  as  the  given  rate  exceeds 
or  falls  short  of  6  per  cent.     (Art.  364.) 

3.  In  finding  tlie  time,  first  determine  the  number  of  entire  calendar 
months ;  then  the  number  of  days  left. 

4.  In  computing  interest,  if  the  mills  are  5  or  more,  it  is  customary 
to  add  I  to  the  cents ;  if  less  than  5,  they  are  disregarded. 

Only  three  decimals  are  retained  in  the  following  Answers,  and 
each  is  found  by  the  rule  under  which  the  Ex.  is  placed. 

2.  What  is  the  amt.  of  1 150. 60  for  i  y.  5  m.  15  d.  at  6%  / 
Analysis. — Int.  of  $1  for  i  y.  5  mos.  on  7  mos.,  equals  .085  ;  15  d. 

it  equals  .0025  ;  and  .085  +  .0025  =  .0875  the  multiplier.     Now  I150.60 
X.0875  — $13  1775,  int.    Finally,  $150.60  + $13. I775=$i63. 7775, ^72.*. 

365.  How  find  the  interest,  when  the  principal,  rate,  and  time  are  given? 
How  find  the  amount,  when  the  principal  and  interest  are  given  ?  liem.  How 
else  is  the  amount  found  ? 


2^H  II^TEREST. 

,5.  Find  the  interest  of  $31.75  for  i  yr.  4  mos.  at  6%. 

4.  What  is  the  int.  of  $49.30  for  6  mos.  24  d.  at  6^  ? 

5.  What  is  the  int.  of  $51.19  for  4  mos.  3  d.  at  y%  ? 

6.  What  is  the  int.  of  I142. 83  for  7  mos.  18  d.  at  5^? 

7.  What  is  the  int.  of  I741.13  for  11  mos.  21  d.  at  6%? 
S.  What  is  the  int.  of  $968.84  for  i  yr.  10  mos.  26d.  at  6%  ? 
9.  What  is  the  int.  of  I639  for  8  mos.  29  d.  at  j%? 

10.  What  is  the  int.  of  $741.13  for  7  mos.  17  d.  at  5^? 
^i.  What  is  the  int.  of  |i 237.63  for  3  mos.  3d.  at  S%? 

12.  What  is  the  int.  of  $2046 J  for  13  mos.  25  d.  at  4%? 

13.  What  is  the  int.  of  $3256.07  for  i  m.  and  3  d.  at  6^? 
"  14.  Find  the  amount  of  $630.37  J  for  9  mos.  15  d.  at  10^? 

15.  Find  the  amount  of  I75.45  for  13  mos.  19  d.  at  7;^? 
J 6.  Find  the  amount  of  $2831.20  for  2  mos.  3  d.  at  g%? 

17.  Find  the  amount  of  $356.81  for  3  m.  11  d.  at  si?c^ 

18.  Find  the  amount  of  $2700  for  4  mos.  3  d.  at  6l-%? 

19.  Required  the  amount  of  $5000  for  ^^  days  at  7^. 

20.  Required  the  int.  of  $12720  for  2  mos.  17  d.  at  4^%. 

21.  What  is  the  amt.  of  $221.42  for  4  mos.  23  d.  at  6%? 

22.  What  is  the  int.  of  $563. 16  for  4  mos.  at  2^  a  month  ? 
Suggestion. — At  2%  a  montli,  the  int.  of  $1  for  4  mos.  is  $.08. 

23.  What  is  the  int.  of  $7216.31  for  3  mos.  at  i^a  month? 

24.  Find  the  int.  of  $9864  for  2  mos.  at  2^%  a  month  ? 

25.  Find  the  amt.  of  $3540  for  17  mos.  10  d.  at  7^%? 

\  26.  What  is  the  interest  on  $650  from  April  17th,  1870, 
to  Feb.  8th,  187 1,  at  6%? 

Analysis. — 1871  y.  2  m.  8  d.    J  Int.  $1  for  9  m.=.o45  $650 

1870  y.  4  m.  17  d.  I    "        "     21  d.  =.0035  .0485 

Time,        o  y.  9  m.  21  d.  j  Multiplier,     .0485       $31,525 

27.  What  is  the  interest  on  $1145  from  July  4th,  1867, 
to  Oct.  3d,  1868,  at  7%. 

28.  What  is  the  interest  on  a  note  of  $568.45  from  May 
2ist,  1861,  to  March  25th,  1862,  at  5^? 

V29.  Required  the  amount  of  $2576.81  from  Jan,  ?»ist, 
1871,  to  Dec.  i8th,  1 87 1,  at  75^ 


INTEREST.  259 


METHOD    BY    ALIQUOT    PARTS. 

366.    To  find   the    Interest   by  Aliquot   Parts,  the   Principal, 
Rate,  and  Time  being  given. 

I.  What  is  the  interest  of  $137  for  3  y.  i  m.  6  d.  at  5^? 

Analysis. — As  tlie  rate  is  5  % ,  tlie  interest  for  S 1 3  7  prin. 

I  year  is  .05  (-[  nu)  of  the  principal,      -        -        -  -05  I'^.te. 

Now$i37x.o5  =  $6.85, I2)$6.85  int.  I  y. 

Again,  the  int.  for  3  y.  is  3  times  as  much  as  for  i  y .,       3  J- 

And  $6.85  (int.  i  y.)  x  3  =  $20.55,         -        -        -  20.55  int.  3  y. 
The  int.  for  i  m.  is  -jV  of  int.  for  i  yr. ;  and 

$6.85^i2  =  .57, 5)        -57   "   ini. 

The  int.  for  6  d.   is   i   of  int.   for   i   m.;   and 

$.57-5  =  . 114, .114'' 6  d. 

Adding  these  partial  interests  together,  we  have         $21,234,  Ans. 
the  answer  required.     Hence,  the 

Rule. — For  one  year. — lluUipli/  the  principal  hy  the 
rate,  expressed  decimally. 

For  tivo  or  more  years. — Multi2)ly  the  interest  for  1  year 
ly  the  number  of  years. 

For  months. — Take  the  aliquot  part  of  i  yearns  interest. 

For  days. — Tahe  the  aliquot  part  of  i  month's  interest. 

That  is,  for  i  m.,  take  -f^  of  the  int.  for  i  y.;  for 
e.  m.,  l\  for  3  m.,  J,  etc. 

For  I  d.,  take  -}q  of  the  int.  for  i  m. ;  for  2  d.,  -^^ ;  for 
6  d.,  I;  for  10  d.,  ^,  etc. 

Note. — This  method  has  the  advantage  of  directness  for  different 
rates;  but  in  practice,  the  preceding  method  is  generally  shorter 
and  more  expeditious. 

2.  What  is  the  int.  of  $143.21  for  2  y.  5  m.  8  d.  at  7^? 

3.  What  is  the  int.  of  $76.10  for  i  y.  3  m.  5  d.  at  6^^? 

366.  How  find  the  interest  for  i  j'ear,  at  any  given  rate,  by  aliquot  parts? 
How  for  2  or  more  years  ?  For  months  ?  For  days  ?  What  part  for  i  month  ? 
for3mo8.?    For6mos.?    For  i  day?  For  5  d.?    For  10  d.?    For  15 d.?   Foraod.? 


4. 


260  IKTEKEST. 

4.  What  is  the  int.  of  $95.31  for  8  m.  20  d.  at  7^? 

5.  What  is  the  int.  of  $110.43  fer  i  yr.  6  m.  10  d.  at  4%? 

6.  What  is  the  int.  of  $258  for  3  yrs.  7  m  at  S%? 

7.  What  is  the  interest  of  $205.38  for  5  yrs.  at  6\%? 

8.  Find  the  interest  of  $361.17  for  11  months  at  8^? 

9.  Find  the  interest  of  $416.84  for  19  days  at  7^^  ? 
10.  At  'j^%,  what  is  the  int.  of  $385.20  for  i  yr.  13  d. 

——II.  At  5^^,  what  is  the  int.  of  $1000  for  i  y.  i  m.  3  d.  ? 

12.  At  8^,  what  is  the  int.  of  I1525.75  for  3  months? 

13.  Eequired  the  interest  of  I12254  for  2 J  years  at  8%? 

14.  AVhat  is  the  amount  of  $20165  ^^^  5  ^^-  ^7  ^-  ^^  7%^ 

METHOD    BY    DAYS. 

367.    To  find  the  Interest  by  DaySf  the  Principal,  Rate, 
and  Time  being  given. 

I.  What  is  the  interest  of  $350  for  78  days  at  6^? 

Analysis. — The  interest  of  $1  at  6^  for  i  day  $350 

is  Jy  of  a  mill  or  ^ifou  dollar.     (Art.  363.)     There-  78 

fore  for  78  days  it  must  be  78  times  BiAyo^t^ooo  "2800 

dol.,  or  ^l.to  times  the  principal.     Again,  since  oAcn 

the  int.  of  $1  for  78  days  is  ^JiTid  times  the  princi-  — ^ — 

pal,  the  interest  of  $350  for  the  same  time  and      o|ooo)$27_^300 
rate  must  be  ^Jusd  times  that  sum.     In  the  opera-         Ans.  $4.55 
tion,  we  multiply  the  principal  by  78,  the  numera- 
tor, and  divide  by  the  denominator  6000.     Hence,  the 

Rule. — Multiply  the  principal  hy  the  numler  of  days,  and 
divide  hy  6000.     Hie  quotient  ivill  he  the  i^iterest  at  6%. 

For  other  rates,  add  to  or  suhfract  from  the  interest  at 
6  per  cent,  such  a  part  of  itself  as  the  required  rate  id 
greater  or  less  than  6%.     (Art.  364.) 

Note. — This  rule,  though  not  strictly  accurate,  is  generally  used 
by  private  bankers  and  money-dealers.  It  is  based  upon  the  suppo- 
sition that  360  days  are  a  year,  which  is  an  error  of  five  days  or  y^ 
of  a  year.  Hence,  the  result  is  7^,  too  larj^e.  When  entire  accuracy 
is  required,  the  result  must  be  diminished  by  7^3-  part  of  itself. 

367.  Eow  compute  interest  by  days  ?    Note.  Upon  what  is  this  method  founded  ? 


INTEREST.  2G1 

2.  A  note  for  $720  was  dated  April  17th,  187 1 :  what  was 
the  interest  on  it  the  i6th  of  the  following  July  at  7^  ? 
Omitting  the  day  of  the  first  date.   (Art.  339,  n.)  $720  prin. 

April  has  30  days— 17  d.=r  13  d.  90  days. 

May     "  31  d.        6!ooo)64|8oo 

June    "  30  d.  6)io.8oint.  6^^. 

July      "  16  d.  1.80    "     ifc. 

Time     "  =90  d.  ^/iS.  $12.60  int.  7^. 

3.  What  is  the  interest  of  $5 1 7  for  ^^  days  at  6%  ? 

4.  What  is  the  interest  of  $208.75  for  6;^  days  at  6%? 

5.  What  is  the  interest  of  I631.15  for  93  days  at  'j%? 

6.  Find  the  interest  of  $1000  for  100  days  at  5^? 

7.  Find  the  amount  of  1 1260. 13  for  120  days  at  6^? 

8.  Kequired  the  interest  of  $3568.17  for  20  days  at  7^? 

9.  Eequired  the  amount  of  $4360.50  for  3  days  at  '/%? 

10.  Find  the  interest  of  $5000  from  May  21st  to  the 
5th  of  Oct.  following  at  6%  ? 

11.  Find  the  interest  of  $6523  from  Aug.  12th  to  the 
5th  of  Jan.  following  at  7;^'? 

12.  Find  the  interest  of  $7510  from  Jan.  5th  to  the 
loth  of  the  following  March,  being  leap  year,  at  6%  ? 

PROBLEM    II. 

368.  To  find  the  Mate,  the  Principal,  the  Interest,  and  the 
Time  being  given. 

I.  A  man  lent  his  neighbor  I360  for  2  yrs.  3  mos.,  and 
received  $48.60  interest:  what  was  the  rate  of  interest? 

Analysis. — The  int.  of  $360  for  $360  x  .01  =  $3.60  int.  i  y. 
I  year  at  I  %  is  $3. 60;  and  for  2  i^y.  3.60  X  2J  =$8.10  "  2\'y. 
$8.10.     Now,  if  $8.10  is  1%  of  the  $8.10)148.60 

principal,  $48  60  must  be  as  many  Ans.  6  per  ct. 

per    cent    as    $8.10  are   contained 
times  in  $48.60,  which  is  6.    Therefore  the  rate  was  6  % .    Hence,  the 

Rule. — Divide  the  given  interest  ly  the  interest  of  the 
principal  for  the  time,  at  i  per  cent. 

36S.  How  find  the  rate,  when  the  principal,  rate,  and  time  are  given  f 


262  Il^TEREST. 

Note. — Sometimes  the  amount  is  mentioned  instead  of  the  prin^ 
cipal,  or  the  intercut.  In  either  case,  the  principal  and  interest  mav 
be  said  to  be  given.  For,  the  amt.=the  prin. +  int. ;  hence,  amt.— 
int,=the  prin. ;  and  amt.— prin.  =:the  int.     (Arts.  365,  loi,  Def.  17.) 

2.  If  $600  yield  1 10.50  interest  in  3  months,  what  is  the 
rate  per  cent  ? 

3.  At  what  rate  will  $1500  pay  me  $52.50  interest 
semi-annually  ? 

4.  At  what  rate  will  $1000  amount  to  $1200  in  3  y.  4  m.  ? 

5.  A  lad  at  the  age  of  14  received  a  legacy  of  I5000, 
which  at  21  amounted  to  $7800 :  what  was  the  rate  of  int.  ? 

6.  A  man  paid  $9600  for  a  house,  and  rented  it  for  $870 
a  year :  what  rate  of  interest  did  he  receive  for  his  money  ? 

7.  At  what  rate  of  int.  will  $500  double  itself  in  1 2  years  ? 

8.  At  what  rate  will  $1000  double  itself  in  20  y.  ?  In  10  y.? 

9.  At  what  rate  will  $1250  double  itself  in  14^  years? 
10.  At  what  rate  must  $3000  be  put  to  double  itself  in 

1 6f  years? 

PROBLEM    III. 

369.  To  find  the  Time,  the   Principal,  the  Interest,  and  the 
Rate  being  given. 

I.  Loaned  a  friend  $250  at  6%,  and  received  $45  interest : 
how  long  did  he  have  the  money  ? 

Analysis.— The  int.  of  $250  at  $250  X  .06  =  $15  int.  for  1  v. 
6%  for  I  yr.  is  $15.     Now  if  $15  $15)845  int. 

int.   require    the  given   principal  ^nsTT  VearS. 

I  yr.  at  6%,  to  earn  $45  int.,  the  '  ^  '' 

same  principal  will  be  required  as  many  years  as  $15  are  contained 
times  in  $45;  and  |45-^$i5=3.  He  therefore  had  the  money  3 
years.     Hence,  the 

Rule. — Divide  tlie  given  interest  hy  tlie  interest  of  the 
principal  for  i  year,  at  the  give7i  rate. 


Note.  How  find  the  principal,  wlien  the  amonnt  and  interest  are  given  ?  How 
find  the  interest,  when  the  amount  and  principal  are  given  ?  369.  How  find  tho 
time,  when  the  principal,  interest,  and  rate  are  given  ? 


IKTEKEST.  363 

Notes.— I.  If  the  quotient  contains  decimals,  reduce  them  to 
months  and  days.     (Art.  294.) 

2.  If  the  amount  is  given  instead  of  the  principal  or  the  internet, 
find  the  part  omitted,  and  proceed  as  above.     (Art.  368.  n.) 

2.  In  what  time  will  $860  amount  to  $989  at  6%  ? 

,  ANALYSIS.-The  amount  $989-$86o=$i29.oo  the  int.,  and  the 
int.  of  $860  for  I  year  at  6^  is  $51.60.  Now  $I29-^$5I.6o=2.5,  or 
2^  years. 

3.  In  what  time  will  $1250  yield  $500,  at  7^  ? 

4.  In  what  time  will  $2200  yield  $100,  at  6%  ? 

5.  In  what  time  will  $10000  yield  $200,  at  %%  ? 

6.  In  what  time  will  $700  double  itself,  at  6%  ? 
Solution.— The  int.  of  $700  for  i  year  at  6%,  is  $42  ;  and  $700^ 

$42  =  i6|  years.  ^n«. 

7.  How  long  must  |i  200  be  loaned  at  7^  to  double  itself? 

8.  In  what  time  will  87500  amount  to  $15000,  at  6%  ? 

9.  In  whd,t  time  will  $10000  amount  to  $25000,  at  8^? 

PROBLEM    IV. 

370.  To  find  the  Principal^  the  Interest,  the  Rate,  and  tho 
Time  being  given. 

1.  What  principal,  at  6%,  will  produce  $60  int.  in  2|  yrs.  ? 

Analysis. — 2^  y.— 30  raos. ;  there/ore  the 

int.  of  $1  for  the  given  time  at  6%  is  15  cents.  2^  y.=r3o  m. 

Now  as  $.15  is  the  int.  of  $1  for  the  given  j^^  of  $  I  =  $11? 

time  and  rate,  $60  must  be  the  int.  of  as  many  ^^. 

dollars  as   $.15   are  contained   times  in   $60;  '   ^l—.    '- 

and  $6o-r-$.i5  =  $4oo,  the  principal  required.  ^^5.  I400  prin. 
Hence,  the 

Rule. — Divide  the  given  interest  hy  the  interest  of%i  for 
t\e  given  time  and  rate,  expressed  decimally. 

2.  At  6fc,  what  principal  will  yield  $100  in  i  year? 

3.  At  7^,  what  principal  will  yield  $105  in  6  months? 

370.  How  find  the  principal,  when  the  interest,  rate,  and  time  arc  given  ♦ 


S64  IKTEEEST. 

4.  What  sum, at  5%,  will  gain  $175  in  i  year  6  months? 

5.  What  sum  must  be  invested  at  6^  to  pay  the  ground 
rent  of  a  house  which  is  $150  per  annum? 

6.  A  gentleman  wished  to  found  a  professorship  with  an 
annual  income  of  I2800 :  what  sum  at  j%  will  produce  it? 

7.  A  man  invested  his  money  in  6%  Government  stocks, 
and  received  $300  semi-annually:  what  was  the  sum 
invested  ? 

8.  A  man  bequeathed  his  wife  $1500  a  year:  what  sum 
must  he  invest  in  6%  stocks,  to  produce  this  annuity  ? 

PROBLEM    V. 

371.  To  find  the  Principal,  the  Amount,  the   Rate,  and  the 
Time  being  giv€n. 

1    I.  What  principal  will  amount  to  I508.20  in  3  y.  at  6%  : 

Analysis.— 3  y.=36  m.  ;    therefore  the 

int.  of  $1  for  the  given  time  at  6%  is  18  cts.,  3  years  =  36  m. 

and  the  anit.r=$i.i8.     (Art.  365.)    Now  as  jnt.  $1=    $.18 

$1.18  is  the  amt.  of  $1  principal  for  the  given  ^.^  %i=i%i  18 

time  and  rate,  $508,20  must  be  the  amount  s»   *  s\<fe      q* 

of  as  many  dollars  principal  as  $1.18  are  con-  ^  *      '^ ! . 

tained  times  in  I508.20  ;  and  $508.20-^$!. 18  Ans.  $430.68 
= $430.68,  the  principal  required.   Hence,  the 

EuLE. — Divide  the  given  amount  hy  the  amount  of%i  for 
the  given  time  and  rate,  expressed  demnally. 

2.  What  principal,  at  6%,  will  amount  to  $250  m  i  year  ? 

3.  What  principal,  at  7^,  will  amouut  to  $356  in 
I  year  3  months  ? 

4.  What  sum  must  a  father  invest  at  6%,  for  a  son  19 
years  old,  that  he  may  have  $10000  when  he  is  21  ? 

5.  What  sum  loaned  at  i^  a  month  will  amount  to 
$1000  in  I  year? 

6.  What  sum  loaned  at  2%  a  month  will  amount  to 
$6252  in  6  mos.  ? 


371.  How  find  the  principal,  when  the  amount,  rate,  and  time  are  p-ivwiT 


PEOMISSOET    NOTES. 

372.  A  JPromissory  Wote  is  a  written  promise  to 
pay  a  ceriain  sum  at  a  specified  time,  or  on  demand. 

Note. — A  Note  should  always  contain  the  words  "value  re 
ceived ;  '*  otherwise,  the  holder  may  be  obliged  to  prove  it  was 
given  f 01-  a  consideration,  in  order  to  collect  it. 

373.  The  person  who  signs  a  note  is  called  the  maker  or  drawer; 
the  person  to  whom  it  is  made  payable,  the  payee;  and  the  person 
who  has  possession  of  it,  the  holder. 

374.  A  Joint  Note  is  one  signed  by  two  or  more  persons. 

375.  The  Face  of  a  Note  is  the  sum  whose  payment  is 
promised.  This  sum  should  be  written  in  words  in  the  body  of  the 
note,  and  m  figures  at  the  top  or  bottom. 

376.  When  a  note  is  to  draw  interest  from  its  date,  it  should 
contain  the  words  "  with  interest ;"  otherwise  no  interest  can  be 
collected.  For  the  same  reason,  when  it  is  to  draw  interest  from  a 
particular  time  after  date,  that  fact  should  be  specified  in  the  note. 

All  notes  are  entitled  to  legal  interest  after  they  become  due, 
whether  they  draw  it  before,  or  not. 

377.  Promissory  Notes  are  of  two  kinds ;  negotiable,  and  non- 
negotiable. 

378.  A  Negotiable  Note  is  a  note  drawn  for  the  payment  of 
money  to  "  order  or  bearer,"  without  any  conditions. 

A  Noit-Neff  of  table  Note  is  one  which  is  not  made  payable  to 
"  order  or  bearer,"  or  is  not  payable  in  money. 

Notes. — i.  A  note  payable  to  A.  B.,  or  "  order,"  is  transferable  by 
indorsement ;  if  to  A.  B,  or  "bearer,"  it  is  transferable  by  delivery. 
Treasury  notes  and  bank  bills  belong  to  this  class. 

2.  If  the  words  "order"  and  "bearer"  are  both  omitted,  the  note 
can  be  collected  only  by  the  party  named  in  it. 

379.  An  Indorser  is  a  person  who  writes  his  name  upgn  the 
back  of  a  note,  as  security  for  its  payment. 

380.  The  Maturity  of  a  Note  is  the  day  it  becomes  legally 
due.    In  most  of  the  United  States  a  note  does  not  become  legally  due 

372.  What  is  a  promissory  note  ?  What  mrticnlar  worflg  Phonld  a  note  con- 
Sain  ?  373.  Wliat  is  the  one  who  signs  it  called  ?  The  one  to  whom  it  is  payable  ? 
The  one  who  has  it  ?  374.  A  joint  note  ?  375.  The  face  of  a  note  ?  377.  Of 
how  many  liinds  are  notes  ?  378.  A  negotiable  note  ?  A  non -negotiable  note  1 
Note.  How  is  the  former  transfwable?  Why  is  the  latter  not  negotiable? 
379.  What  is  an  indorser  ?     380.  The  maturity  of  a  note  ?    • 

12 


2Q6  ll^TEREST. 

until  three  days  after  the  time  specified,    Tliese  three  days  are  called 
dai/s  of  grace.    Hence,  a  note  matures  on  the  last  day  of  grace. 

38X.  When  a  note  is  given  for  any  number  of  months,  calendar 
moauis  are  always  to  be  understood. 

382.  If  a  note  is  payable  on  demand,  it  is  legally  due  as  soon  as 
presented.  If  no  time  is  specified  for  the  payment,  it  is  understood 
to  be  on  demand. 

383.  A  Protest  is  a  written  declaration  made  by  a  notary 
public,  that  a  note  has  been  duly  presented  to  the  maker,  and  ha& 
not  been  paid. 

Note.-— A  protest  must  be  made  out  the  day  the  note  or  draft 
matures,  and  sent  to  the  indorser  immediately,  to  hold  hira  respoiuibls. 


ANNUAL   INTEREST. 

384.  When  notes  are  made  "  with  interest  payable  annually," 
some  States  allow  simple  legal  interest  on  each  year's  interest  from 
the  time  it  becomes  due  to  the  time  of  final  settlement. 

385.  To  compute  Annual  Interest,  the  Principal,  Rate, 
and  Time  being  given. 

1.  What  is  the  amount  due  on  a  note  of  $500,  at  6^, 
in  3  years  with  interest  payable  annually  ? 

Solution.— Principal,  $500.00 

Int.  for  I  y.  is  $30  :  for  3  y.  it  is  $30  x  3,  or  90.00 

Int.  on  ist  annual  int.  for  2  y.  is  9.60 

2d       "         "      "    I  y.  is  1.80 

The  Amt.  is  $595  40-     Hence,  the  Ans.  $595.40 

KuLE. — Find  the  interest  on  the  principal  for  the  given 

time  and  rate;  also  find  the  simple  legal  interest  on  each 

yearns  interest  for  the  ti7ne  it  has  remained  unpaid. 

Tlie  sum  of  the  p)fincipal  and  its  interest,  ivith  the 
interest  on  the  unpaid  interests,  will  he  the  amount. 

2.  What  is  the  amount  of  a  note  of  $1000  payable  in 
4  years,  with  interest  annually,  at  7^?      Ans.  $1309.40 

382.  When  is  a  note  on  demand  due?  383.  What  is  a  protest?  385.  How 
computo  annual  interest  ? 


PARTIAL    PAYMENTS. 

386.  JPartial  JPaymenfs  are  parts  of  a  debt  paid 
at  diflFerent  times.  The  suras  paid,  with  the  date,  are 
usually  written  on  the  back  of  the  note  or  other  obliga- 
tion, and  are  thence  called  indorsements. 

UNITED    STATES    RULE. 

387.  To  compute  Interest  on  Notes  and  Bonds,  when  Partial 
Payments  have  been  made. 

1.  Fi7id  the  amount  of  tlie  prmcijoal  to  the  time  of  the 
first  loayment.  If  the  payment  equals  or  exceeds  the  interest, 
subtract  it  from  this  amount,  and^  considering  the  remainder 
a  new  'principal,  proceed  as  before. 

II.  If  the  payment  is  less  than  the  interest,  find  the 
amount  of  the  same  principal  to  the  next  payment,  or  to 
the  period  when  the  sum  of  the  payments  equals  or  exceeds 
the  interest  then  due,  and  subtract  the  sum  of  the  payments 
from  this  amount. 

Proceed  in  this  manner  with   the  balance  to  the  time 

of  settlement. 

Notes. — i.  This  metliod  was  early  inaugurated  by  the  Suprem* 
Court  of  the  United  States,  and  is  adopted  by  New  York,  Massa^ 
chusetts,  and  most  of  the  States  of  the  Union. — Chancellor  Kent. 

2.  The  following  examples  show  the  common  forms  of  promissory 
notes.  1\\G  first  is  negotiable  by  indm'sement ;  the  second  by  transfer i 
the  third  is  &  joint  note,  but  not  negotiable. 


$750- 

Washington,  D.  C,  Jan.  jth,  1870. 

I.  Four  months  after  date,  I  promise  to  pay  to  the  ordei 

of  George  Green,  seven  hundred  and  fifty  dollars,  with 

interest  at  6fc,  for  value  received. 

Hen^rt  Bkowij". 

386.  What  are  partial  payments  ?    387.  WTiat  is  the  United  States  mlc  ? 


268  INTEREST. 

On  this  note  the  following  payments  were  indorsed. 

June  loth,  1870,  $43.     Feb.  17th,  187 1,  $15.45-  ^"o^-  23d, 
187 1,  $78.60.     What  was  due  Aug.  25th,  1872  ? 

OPERATION. 

Principal,  dated  Jan.  7th,  1870,  $750.00 

Int.  to  ist  payt.  June  lotli,  1870  (5  m.  3  d.)    (Art.  365),  ^9-^3 

Amount,  —  769-13 

ist  payment  June  loth,  1870,  43'^^ 

Remainder  or  new  principal,  =726.13 

Int.  from  ist  payt.  to  Feb.  I7tli,  1871  (8  m.  7  d.),  29.89 
2I  payt.  less  than  int.  due,                                       $I545 

Int.  on  same  prin.  to  Nov.  23d,  1871  (9  m.  6  d.),  3340 

Amount,  =  789.42 

3d  payt.  to  be  added  to  2d,                                     $7^-6o  =    94-05 

Remainder  or  new  principal,  =  695.37 

Int.  to  Aug.  25th,  1872  (9  m.  2  d.),  S^-S^ 

Balance  due  Aug.  25tli,  1872,  =  $726.89 


$1500. 

New  Orleans,  July  1st,  1869. 

2.  Two  years  after  date,  we  promise  to  pay  to  James 

'^TFnderhill  or  bearer,  fifteen  hundred  dollars,  with  interest 

at  7^,  value  received. 

G.  H.  Dennis  &  Co. 

Indorsements: — Received,  Jan.  5th,  1870,  $68.50.    Aug. 

8t]i,  1870,   $20.10.     Feb.  nth,  187 1,  $100.     How  much 

was  due  at  its  maturity  ? 


$930- 


St.  Louis,  March  st7i,  i860. 
3.  On  demand,  we  jointly  and  severally  promise  to  pay 
J.  C.   Williams,  nine  hundred   and   thirty  dollars,  with 
interest  at  8;^,  value  received.  Thomas  Benton. 

Henry  Valentine. 
Indorsements: — Received   Oct.  loth,  i860,  $20.    'Nov. 
i6th,   1861,  $250.13.     June  20th,  1862,  $310:  what  was 
ine  Jan.  30th,  1863? 


PARTIAL     PAYMBlfTS.  J&69 


MERCANTILE    METHOD. 

388.  When  notes  and  interest  accounts  payable  within  a 
year,  receive  partial  payments,  business  men  commonly 
employ  the  following  method : 

Find  the  amount  of  the  whole  deU  to  the  time  of  settle- 
ment; also  find  the  amount  of  each  payment  from  the  time 
it  was  made  to  the  time  of  settlements 

Subtract  the  amount  of  the  payments  from  the  amount  of 
the  debt;  the  remainder  luill  be  the  balance  due. 

4.  A  debt  of  $720.75  was  due  March  15th,  1870,  on 
which  the  following  payments  were  made :  AprQ  3d,  $170 ; 
May  20th,  $245.30;  June  17th,  $87.50.  How  much  was 
due  at  6%,  Sept.  5th,  1870  ? 

Principal  dated  March  15th,  1870,  $720.75 

Int.  to  settlement  (174  d.)  =  $720.75  x  .029,    (Art.  367.)  20.90 

Amount,  Sept.  5,  '70,  ~  74i'65 

ist  payt.,  $170.     Time,  155  d.     Amt ,  ==  $174.39 

2d  payt.,  $245.30.     Time,  108  d.     Amt.,       =     249.72 
3d  payt.,  $87.50.     Time,  Sod.     Amt.,  =        88.67 

Amt.  of  tlie  Payts.,  =  512.78 

Balance  due  Sept.  5th,  1870,  $228.87 

5.  Sold  goods  amounting  to  $650,  to  be  paid  Jan.  ist, 
1868.  June  loth,  received  $125;  Sept.  13th,  $75.50;  Oct. 
3d,  $210:  what  was  due  Dec.  31st,  1868,  at  6%  interest? 

6.  A  note  for  $820,  dated  July  5  th,  1865,  payable  in 
I  year,  at  7^  interest,  bore  the  following  indorsements : 
Jan.  loth,  1866,  received  $150;  March  20th,  received 
$73.10;  May  5th,  received  $116;  June  15th,  received 
$141.50:  what  was  due  at  its  maturity? 

7.  An  account  of  $1100  due  March  3d,  received  the 
following  payments:  June  ist,  $310;  Aug.  7th,  $119; 
Oct.  17th,  $200:  what  was  due  on  the  27th  of  the  next 
Dec,  allowing  7^  interest  ? 


S70  IKTEBEST. 


CONNECTICUT    METHOD. 

389.  I.  When  the  first  payment  is  a  year  or  more  from 
the  time  the  interest  commenced. 

Find  the  amount  of  the  principal  to  that  time.  If  the 
payment  equals  or  exceeds  the  interest  due,  subtract  it  from 
the  amount  thus  found ^  and  consideriyig  the  remainder  a  new 
'principal,  proceed  thus  till  all  the  j)ayments  are  absorbed. 

11.  When  a  payment  is  made  before  a  year's  interest  has 
accrued. 

Firid  the  amount  of  the  principal  for  i  year ;  also  if  the 
payment  equals  or  exceeds  the  interest  due,  find  its  amount 
from  the  time  it  was  made  to  the  end  of  the  year,  and  sub- 
tract this  amount  from  the  amount  of  the  principal;  and 
treat  the  remainder  as  a  neio  principal. 

But  if  the  payment  be  less  than  the  interest,  subtract  the 
payment  only,  from  the  amou7it  of  the  principal  thus  found, 
and  proceed  as  before. 

Note. — If  the  settlement  is  made  in  less  than  a  year,  find  the 
amount  of  the  principal  to  the  time  of  settlement;  also  find  the 
amount  of  the  payments  made  during  this  period  to  the  same  date, 
and  subtracting  this  amount  from  that  of  the  principal,  the  remainder 
will  be  the  balance  due. — Kirby'8  Beports. 


$650. 

New  Haven,  April  12th,  i860. 

8.  On  demand,  I  promise  to  pay  to  the  order  of  George 

Selden,  six  hundred  and  fifty  dollars,  with  interest,  value 

received.  Thomas  Sawyer. 

Indorsements: — May  ist,  186 1,  received  $116.20.  Feb. 
Toth,  1862,  received  $61.50.  Dec.  12th,  1862,  received 
$12.10.  June  20th,  1863,  received  $110:  what  was  due 
Oct.  2ist,  1863? 

388.  What  is  the  mercantile  method  ?    389.  What  is  tht  Connecticnt  method  ? 


PARTIAL     PAYMENTS.  271 

Principal,  dated  April  12th,  i860,        '  $650.00 

lut.  to  ist  payt.  May  ist,  1861  (i  y.  19  d.),  41.06 

Amount,  May  i,  '61,  =  691.06 

ist  payt.  May  ist,  1861,  11 6. 20 

Bemainder  or  New  Prin.,  May  i,  '61,  =  574-86 

Int.  to  May  i,  '62,  or  i  y.  (2d  paj't.  being  short  of  i  y.),  34*49 

Amount,  May  i,  '62,  =  609.35 

A-mt.  of  2d  payt.  to  May  i,  '62  (2  m.  19  d.),  62.31 

Rem.  or  New  Prin.,  May  i,  '62,  =  5 47 -04 

Amt.,  May  i,  '63  (i  y.),  =  579-86 

3d  payt.  (being  less  than  int.  due),  draws  no  int.,  12.10 

Rem.  or  New  Prin.,  May  i,  '63,  =  S^l'l^ 

Amt.,  Oct.  21,  '63  (5  m,  20  d.),  =  583-85 

Amt.  of  last  payt.  to  settlement  (4  m.  id.),  =  112.22 

Balance  due  Oct.  21,  63,  =  ^471.63 

Note. — For  additional  exercises  in  the  Connecticut  rule,  the 
(Student  is  referred  to  Art.  387. 

VERMONT    RULE. 

390.  I-  "  When  payments  are  made  on  notes,  bills,  or  similar 
obligations,  whether  payable  on  demand  or  alra  sDecified  time, '  with 
interest,'  sucli  payments  shall  be  applied;  First,  to  liquidate  the 
interest  that  has  accrued  at  the  time  of  such  payments ;  and, 
secondly,  to  the  extinguishment  of  the  principal.'* 

II.  "  The  annual  interests  that  shall  remain  unpaid  on  notes,  bills. 
or  similar  obligations,  whether  payable  on  demand  at  a  specified 
time,  "  with  interest  annually,"  shall  be  subject  to  simple  interest 
from  the  time  they  become  due  to  the  time  of  final  settlement." 

III.  "  If  payments  have  been  made  in  any  year,  reckoning  from  the 
time  such  annual  interest  began  to  accrue,  the  amount  of  such  pay- 
ments at  the  end  of  such  year,  with  interest  thereon  from  the  time 
of  payment,  shall  be  applied  ;  Mrst,  to  liquidate  the  simple  interest 
that  has  accrued  from  the  unpaid  annual  interests. 

"  Secondly,  To  liquidate  the  annual  interests  that  have  become  due 
"  Thirdly,  To  the  extinguishment  of  the  princijml." 

391.  The  Rule  of  New  Hampshire,  when  partial  paymen  s  are 
made  on  notes  "  with  interest  annually,"  is  essentially  the  same  as 
the  preceding.  But  "  where  payments  are  made  expressly  on  account 
of  interest  accruing,  but  not  then  due,  they  are  applied  when  t^e 
interest  falls  due,  without  interest  on  such  payments." 


%7t  PARTIAL     PAYMENTS. 


$1500. 

BUKIilNGTON,  Feb.  \9^,  xHi<.. 

9.  On  demand,  I  promise  to  pay  to  the  order  of  Jarccl 
Sparks,  fifteen  hundred  dollars,  with  interest  annually, 
Talue  received.  Augustus  Waeeen. 

Indorsements : — Aug.  i  st,  1 865,  received  $  1 60 ;  July  1 2th, 
1866,  $125;  June  i8th,  1867,  $50.  Required  the  amouri 
due  Feb.  ist,  1868. 

Principal,  $1500.00 

lut.  to  Feb.  I,  '66  (i  jr.  at  6%),  90.00 

Amount,  —  i590.0(?; 

1st  payment  Aug.  i,  '65,  $160.00 

Int.  on  same  to  Feb.  i,  '66  (6  mos.),  4.80                164.80 

Remainder  or  new  principal,  1425.2(2 

Int.  on  same  to  Feb.  i,  '67  (i  yr.),  ^5-5  ' 

Amount,  =  15 10.7 1 

2d  payment  July  12,  '66,  $125.00 

Int.  on  same  to  Feb.  i,  '67  (6  m.  20  d.),  4.16                129.16 

Remainder  or  new  principal,  =  1381.55 

Int.  on  same  to  Feb.  i,  '68  (i  yr.),  82.89 

Amount,  =  1464.44 

3d  payment  June  18,  '67,  $50.00 

Int.  on  same  to  Feb.  i,  '68  (7  m.  14  d.),  1.87                  5 1 '87 

Bal.  Feb.  ist,  1868,  =  $1412.57 

$2000. 

•  Concord,  Jan.  isth,  1869. 

10.  Two  years  after  date,  I  promise  to  pay  to  the  order 
of  Lewis  Hunt,  two  thousand  dollars,  "with  interest i** 
the  payee,  Jan.  15th,  1870,  received,  by  agreement,  $200 
on  account  of  interest  then  accruing.  What  was  due  oq 
this  note  Jan.  15th,  1871,  by  the  Vt.  and  N.  H.  rules? 

By  the  Vt.  rule,  tlie  bal.=$2240— $2i2=:$2028  )    . 
"       N.H.  rule,     "       =  $2240- $200= $2040  f       ** 

In  the  former,  interest  is  allowed  on  the  payment  from  its  date  to 
ttie  settlement :  in  the  latter,  it  is  not 


COMPOUND    lE'TEEEST. 

392.  Interest  may  be  compounded  annually,  semi^ 
annually,  quarterly,  or  for  any  other  period  at  which  the 
interest  is  made  payable. 

393.  To  compute   Comxxntnd  Interest,  the   Principal, 
the  Rate,  and  Period  of  compounding  being  given. 

I.  What  is  the  compound  interest  of  $600  for  3  years, 
at  6^? 

Principal,  $600.00 

Int.  for  ist  year,  $600  x  .06,  3^-^^ 

Amt.  for  I  y.,  or  2d  prin.,  =636.00 

Int.  for  2d  year,  $636  x  .06,  3^-^^ 

Amt.  for  2  yrs.,  or  3d  prin.,  ==  674.16 

Int.  for  3d  year,  $674.16  x  .06,  4^45 

Amt.  for  3  years,  =  714.61 

Original  principal  to  be  subtracted,  600.00 

Compound  int.  for  3  years,  =  $1 14.61 

Hence,  the 

EuLE.— I.  Fi?id  the  amount  of  the  principal  for  the  first 
veriod.  Treat  this  amount  as  a  new  pt'fi^t^cipal,  and.  find 
the  amount  due  on  it  for  another  period,  and  so  on  through 
every  period  of  the  given  time. 

II.  Subtract  the  given  priyicipal  from  the  last  amount, 
and  the  remainder  zuill  be  the  c07npound  interest. 

Notes. — i.  If  there  are  months  or  days  after  tlie  last  regular 
period  at  which  the  interest  is  compounded,  find  the  interest  on  the 
amount  last  obtained  for  them,  and  add  it  to  the  same,  before  sub- 
tracting the  principal. 

2.  Compound  "interest  cannot  be  collected  \)^  law;  but  a  creditor 
may  receive  it,  without  incurring  the  penalty  of  usury.  Savingi 
Banks  pay  it  to  all  depositors  who  do  not  draw  their  interest 
when  due.  .^ 

\r,.,^--2.  What  is  the  compound  int.  of  $500  for  3  yrs.  at  7^?   //^•^ 

3.  What  is  the  compound  int.  of  $750  for  4  yrs.  at  5^?   f  /  t 

393.  How  find  compound  interoat,  when  the  principal,  rate,  and  time  are  givMi? 


> 


274 


COMPOUJs^D     INTEREST. 


4.  What  is  the  com.  int.  of  $1000  for  2  y.  7  m.  9  d.  at  6^? 

5.  What  is  the  interest  of  $1360  for  2   years  at  7^, 
compounded  semi-annually  ? 

6.  What  is  the  amount  of  $2000  for  2  years  at  4%,  com- 
pounded quarterly  ? 

COMPOUND     INTEREST    TABLE. 

Showing  the  amount  of  %i,  at  3,  4,  5,  6,  and  7;;^  com- 
pound  interest,  for  any  number  of  years  from  i  to  25. 


Yrs. 
I. 

3%. 

4%. 

5%. 

tfo. 

r/e. 

1.030000 

1.040000 

1.050000 

1.060000 

1.07  000 

2. 

1.060900 

I.081  600 

1. 102  500 

1. 123  600 

1. 14  490 

3- 

1.092  727 

1. 124  864 

1. 157  625 

1. 191  016 

1.22  504 

4 

1. 125  509 

1. 169  859 

1.215  506 

1.262477 

I.3IO79 

5- 

1. 159  274 

1. 216  653 

1.276  282 

1.338226 

1.40255 

6. 

1. 194  052 

1.265  319 

1.340096 

I.4I85I9 

1.50073 

7- 

1.229  ^74 

1-315932 

1.407  100 

1.503630 

1.60578 

8. 

1.266  770 

1.368569 

1-477  455 

1.593848 

I.71818 

9- 

1-304773 

1.423  312 

1-551328 

1.689479 

1.83  845 

10. 

1-343916 

1.480  244 

1.628895 

1.790848 

1.96  715 

II. 

1.384234 

1-539  451 

1.710339 

1.898  299 

2.10485 

12. 

1.425  761 

1.601  032 

1-795  856 

2.012  196 

2.25  219 

13- 

1.468534 

1.665  074 

1.885  649 

2.132  928 

2.40  984 

14. 

1.512590 

1.731676 

1.979932 

2.260  904 

2-57853 

15- 

1.557967 

1.800  944 

2.078  928 

2-396558 

2.75  903 

16. 

1.604  706 

1.872  981 

2.182875 

2.540352 

2.95  216 

17. 

1.652848 

1.947  900 

2.292  018 

2.692773 

3.15  881 

18. 

1.702433 

2.025  817 

2.406  619 

2-854339 

3-37  293 

19. 

1-753506 

2.106  849 

2.526950 

3.025*600 

3.61652 

20. 

1.806  III 

2.191  123 

2.653  298 

3-207 135 

3.86  968 

21. 

1.860295 

2.278768 

2-785  963 

3-399  564 

4.14056 

22. 

1.916  103 

2.369919 

2.925  261 

3-603537 

4.43  040 

23- 

1-973587 

2.464  716 

3-071  524 

3-819750 

4-74052 

24. 

2.032  794 

2.563304 

3.225  100 

4-048  935 

5.07  236 

25- 

2.093  778 

2.665  ^Z^ 

3.386355 

4.291  871 

5-42  743 

COMPOUND     INTEREST.  '^75 

394.   To  find  Compound  Interest  by  the  Table. 

^       7.  Yfhat  is  the  amount  of  $900  for  6  yrs.  at  7;?,  the  int 
^    compounded  annually  ?    A¥hat  is  the  compound  interest  ? 

Solution — Tabular  amt.  of  $1  for  6  yrs.  at  75^,  $1.50073 

The  principal,  9°° 

Amt.  for  6  yrs.,  =  $1350.65700 
The  principal  to  be  subtracted  from  amt.,  900 

Compoundlni.ioT  6  yv^.,  =     $450,657 
Hence,  the 

Rule. — I.  Multiply  the  tabular  amount  of%ifor  the  given 
time  and  rate  hy  the  principal;  the  product  tvill  he  the 
amount. 

II.  From  the  amount  subtract  the  principal,  and  ilie 
remainder  will  be  the  compound  interest. 

Note. — If  the  given  number  of  years  exceed  that  in  the  Table, 
find  the  amount  for  any  convenient  period,  as  half  the  given  years  ; 
then  on  this  amount  for  the  remaining  period. 

8.  What  is  the  interest  of  $800  for  9  years  at  6'^,  com- 
pounded annually  ? 

9.  What  is  the  int.  of  $1100  for  12  years  at  7;^,  com- 
pounded annually  ? 

10.  What  is  the  int.  of  $1305  for  16  years  at  5^,  com- 
pounded annually? 

11.  What  is  the  amount  of  $4500  for  15  years  at  4^, 
compounded  annually? 

12.  What  is  the  amount  of  $6000  for  25  years  at  7;^, 
compounded  annually  ? 

13.  What  is  the  amount  of  $3800  for  30  years  at  6^, 
compound  interest  ? 

14.  What  is  the  compound  interest  of  $4240  at  5%  for 
40  years  ? 

15.  What  is  the  amount  of  $1280  for  50  years  at  7^ 
compound  interest  ? 

16.  What  will  $100  amount  to  in  60  years  at  6%  com- 
pound interest  ? 


276  DISCOUKT. 

DISCOUNT. 

395.  Commercial  Discount  is  a  deduction  of  a 
certain  per  cent  from  the  price-list  of  goods,  the  face  of 
bills,  &c.,  without  regard  to  time. 

396.  True  Discount  is  the  difference  between  a 
debt  bearing  no  interest  and  \i^  present  worth. 

397.  The  l^resetit  WortJi  of  a  debt  payable  at  a 
future  time  without  interest,  is  the  sum,  which,  put  at 
legal  interest  for  the  given  time,  will  amount  to  the  debt. 

397,  a.  To  find  the   Net  Proceeds  of  Commercial   Discount. 

1.  Sold  goods  marked  $1560^  a-t  20^  discount,  on  4  m., 
then  deducted  $%  for  cash.    Required  the  net  proceeds? 

Analysis.  —  $1560  x  .20  =  $312.00,  and  $1560  —  $312  =  $1248. 
$1248  X  .05 =$62.40,  and  $1248— $62.4o=$ii85.6o,  net.     Hence,  the 

EuLE. — Deduct  the  commercial  discount  from  the  list 
price,  and  from  the  remainder  tahe  the  cash  discount. 

2.  What  is  the  net  value  of  a  bill  of  $3500,  at  15^  dis- 
count, and  5^  additional  for  cash?  Ans.  $2826.25. 

3.  What  is  the  net  value  of  a  bill  amounting  to  I5280, 
at  1 2 i^  discount  ?  A71S.  I4620. 

398.  To  find  the  Frese^it  Worth  of  a  debt,  the  Rate  and 

Time  being  given. 

1.  What  is  the  present  worth  of  $250.51,  payable  in 
8  months  without  interest,  money  being  worth  6%  ? 

Analysis. — The  amount  of   $1   for  8  $i,o4==:amt.  I2 

mos.,  at  6%,  is  $1.04;  therefore  $1  is  the  $1.04)1250.5 1  debt 
present  worth  of  $1.04,   due  in   8  mos.,  $240  871;    Ans 

and$25o.5i^$i.04=$240.875.  Hence, the 

Rule. — Divide  the  debt  hy  the  amount  of  %i  for  the  given 
time  and  rate  ;  the  quotient  will  he  the  present  loorth. 

2.  Find  the  present  worth  of  $300,  due  in  10  m.,  Avhen 
interest  is  7^. 

395.  What  is  discount?  396.  Commercial?  True?  397.  Present  worth? 
397,  a.  How  And  net  proceeds  of  commercial  discount?    398.  Present  worth  ? 


Discoui^T.  277 

3.  Find  the  present  worth  of  $500,  due  in  i  year,  when 
interest  is  8^. 

4.  What  is  the  present  worth  of  a  note  for  $1250,  pay- 
able in  6  mos.,  interest  being  6%  ? 

5.  A  man  sold  his  farm  for  $2500  on  i  year  without 
interest:  what  is  the  present  worth  of  the  debt,  money 
being  7^? 

6.  What  is  the  present  worth  of  a  legacy  of  $5000, 
payable  in  2  years,  when  interest  is  6%  ? 

399.     To    find    the    True   Discount,   the    Rate    and    Time 
being   given. 

7.  What  is  the  true  discount  at  6^0  on  a  note  of  $474.03 
due  in  6  months  and  3  days  ? 

ANALYSis.-The    amount    of   $1    for  %i.o^ok)%a.a.ox 

6  mos.  and  3  days  is  $1.0305.     (Art.  365.)  ^      --^?~- 

Therefore,  the  present  worth  of  the  note  ^40° 

is  $474.03 -r- 1.0305,  or  $460.     (Art.  398.)  $474.03— 46o  =  $i4.03 
But,  by  definition,  the  debt,  minus  the 

present  worth,  is  the   true  discount  ;   and   $474.03— $460  (present 
worth)=$i4,o3,  the  Ans.     Hence,  the 

EuLE. — First  find  the  present  worth;  then  suUrad  it 
from  the  deht. 

8.  Find  the  discount  on  I2560,  due  in  7  months,  at  6%. 

9.  Find  the  discount  on  $2819,  due  in  9  months,  at  <,%. 

10.  Bought  $2375  worth  of  goods  on  6  months:  what  is 
the  present  worth  of  the  bill,  at  8^  ?  The  discount  ? 

11.  A.t  6%  what  is  the  present  worth  of  a  debt  of  $3860, 
half  of  which  is  due  in  3  months  and  half  in  6  months  ? 

12.  What  is  the  difference  between  the  int.  of  $6000  for 
I  y.  at  6%,  and  the  discount  for  i  y.  at  6%  ? 

13.  Bought  a  house  for  $5560  on  i-J  year  vithout 
interest :  what  would  be  the  discount  at  7^,  if  paid  down  ? 

14.  If  money  is  worth  7^,  which  is  preferable,  $15000 
cash,  or  $16000  payable  in  a  year  without  interest? 

•^99,  How  find  the  true  discount,  when  the  rate  and  time  are  given  ? 


[ '  (yx  eAM^^  y    \  ^  -^^ 


278  BANKS    AI^D    BAKK     DISCO  U2hT. 

BANKS     AND     BANK     DISCOUNT. 

400.  JSanJcs  are  incorporated  iustitutions  which  deal 
in  money.  They  are  of  four  kinds:  banks  of  Dejjosit, 
Discount^  Circulation,  and  Savings. 

401.  A  J^anh  of  Deposit  is  one  that  receiyes 
money  for  safe  keeping,  subject  to  the  order  of  the 
depositor. 

A  Hanh  of  Discount  is  one  that  loans  money, 
discounts  notes,  drafts,  etc. 

A  Danh  of  Circulation  is  one  that  makes  and 
issues  hills,  which  it  promises  to  pay,  on  demand. 

A  Savings  Sank  is  one  that  receives  small  sums 
on  deposit,  and  puts  them  at  interest,  for  the  benefit 
of  depositors. 

402.  Dank  Discount  is   simple  interest  paid  in 

advance. 

Notes. — i.  A  note  or  draft  is  said  to  bo  discounted  wlien  the 
interest  for  the  given  time  and  rate  is  deducted  from  tlie  face  of  it, 
and  the  halance  paid  to  the  holder. 

2.  The  part  paid  to  the  holder  is  called  the  proceeds  or  avails  of 
the  note;  the  part  deducted,  the  discount. 

3.  The  time  from  the  date  when  a  note  is  discounted  to  its  ma- 
turity, is  often  called  the  Term  of  Disco  ant, 

403.  To  find  the  Bank  Disco^mt,  the  Face  of  a  Note,  the 
Time,  and  the  Rate  being  given. 

I.  What  is  the  bank  discount  on  $450  for  4  m.  at  6;^? 

Ana-lysts. — The  int.  of  $1  for  4  m.  3  d.  is  $.0205  ;  and  $450  x  .0205 
—$9,225,  the  bank  discount  required.  (Art.  380.)     Hence,  the 

Rule. — Compute  the  interest  on  the  face  of  the  note  at 
the  given  rate,  for  3  days  more  than  the  given  time. 

To  find  the  proceeds : — Suhtract  the  discount  from  the 
face  of  the  note, 

400.  What  is  a  bank?  401.  A  bank  of  deposit?  Discount?  Circulation? 
Bavingsbank?  402.  Wiiat  is  bank  discount  ?  403.  How  find  tlie  bank  discount, 
when  the  face  of  a  note,  the  rate  and  time  are  given  ?    IIow  find  the  proceeds  ? 


BANKS     AND     BAKK     DISCOUKT.  579 

Note. — If  a  note  is  on  interest,  find  its  amount  at  maturity,  and 
taking  this  as  tlie/«ce  of  the  note,  cast  the  interest  on  it  as  above. 

2.  Find  the  term  of  discount  ainl  proceeds  of  a  note  for 
$500,  on  90  d.,  dated  June  5th,  1873,  nnd  discounted  July 
3d,  1873,  at  ']%.  Ans.  65  d.;  Pro.,  I493.68. 

3.  Find  the  proceeds  of  a  note  of  $730  due  in  3m.  at  6%. 

4.  Find  the  proceeds  of  a  draft  for  $1000  on  60  d.  at  6%. 

5.  Find  the  maturity,  term  of  discount,  and  proceeds  of 
a  note  of  $1740,  on  6m.,  dated  May  i,  '73,  and  dis.  Aug.  21st, 
'73? at  5^.     Ans.  Nov.  4th,  '73 ;  Time,  75  d. ;  Proceeds,  — . 

6.  What  is  the  difference  between  the  true  and  bank 
discount  on  $5000  due  in  lyear  at  6%? 

yr-  7.  A  jobber  buying  $7^00  worth  of  goods  for  cash,  sold 
them  on  ^  mos.  at  1 2-^%  advance,  and  got  the  note  dis- 
counted at  y/o  to  pay  the  bill :  how  muoh  did  he  make  ? 

404.  To  find  the  Face  of  a  Wote,  that  the  proceeds  at  Bank 
Discount  shall  be  a  specified  sum,  the  R.  and  T.  being  given. 

8.  For  what  sum  must  a  note  be  drawn  on  6  mos.  that 
at  6%  bank  discount,  the  proceeds  shall  be  $500  ? 

Analysis. — The  bank  discount  of  $1  for  6  mos.  3  d.  at  6%  is 
$.0305 ;  consequently  the  proceeds  are  $.9695.  Nuw  as  §.9695  are 
the  proceeds  of  Si,  $500  must  be  the  proceeds  of  as  many  dollars  as 
$.9695  is  contained  times  in  $500;  and  $5oo^$.9695  =  $5i5.729,  the 
face  of  the  note.     Hence,  the 

EuLE. — Dioide  the  rjivoi  j^i^oceeds  1)])  the  proceeds  of  %\ 
for  the  given  time  and  rate. 

9.  What  must  be  the  face  of  a  note  on  4  mos.  that  when 
discounted  at  ']%  the  proceeds  may  be  $750  ? 

10.  What  was  the  face  of  a  note  on  60  days,  the  pro- 
ceeds of  which  being  discounted  at  5^,  were  I1565  ? 

11.  If  the  avails  are  $2165.45,  the  time  4  mos.,  and  the 
rate  of  bank  discount  8;:^,  what  must  be  the  face  of  the  note? 

12.  Bought  a  house  for  $7350  cash :  how  large  a  note  on 
4  mos.  must  I  have  discounted  at  bank  6%  to  pay  this  sum? 


404.  How  And  how  large  to  make  a  note  to  raise  a  Bpecifled  sura,  when  the  rate 
ftfid  time  are  ifhrcn  ? 


2B0  STOCK    INVESTMEiJ^TS. 

STOCK    INVESTMENTS. 

405.  Stocks  are  tlie  funds  or  capital  of  incorporated 
companies. 

An  Incorporated  Cofnpany  is  an  association 
authorized  by  law,  to  transact  business. 

Note. — Stocks  are  divided  into  equal  parts  called  shares,  and  tlie 
owners  of  tlie  shares  are  called  stockholders.  These  shares  vary  from 
$25  to  $500  or  $1000.  They  are  commonly  $100  each,  and  will  be 
BO  considered  in  the  following  exercises,  unless  otherwise  stated. 

406.  Certificates  of  Stoclz  are  written  statements, 
specifying  the  number  of  shares  to  which  holders  are 
entitled.     They  are  often  called  scrip. 

407.  The  Var  value  of  stock  is  the  sum  named  on 
the  face  of  the  scr'.p,  and  is  thence  called  its  nominal  value. 

The  Market  value  is  the  sum  for  whicli  it  sells.    • 
Notes. — i.  When  shares  sell  for  their  nominal  value,  they  are  at 

•par ;  when  they  sell  for  riiore,  they  are  ahove  par,  or  at  a  premium  ; 

when  they  sell  for  less,  they  are  helow  par,  or  at  a  discount. 

2.  When  stocks  sell  at  par,  they  are  often  quoted  at  100  ;  when  8  % 

ahov:e  par,  at  108  ;  when  8  %  'below  par,  at  92.     The  term  par,  Latin, 

signifies  equal ;  hence,  to  be  at  par,  is  to  be  on  an  equality. 

408.  The  Gross  Eai'nings  of  a  Company  are  its  entire 

receipts. 

The  Net  Earnings  are  the  sums  left  after  deducting  all 
expenses. 

409.  Instalments  are  portions  of  the  capital  paid  by  the 
stockholders  at  diflPerent  times. 

410.  Dividends  are  portions  of  the  earnings  distributed  among 
the  stockholders.  They  are  usually  made  at  stated  periods ;  as, 
annually ;  semi-annually,  etc. 

411.  A  Hond  is  a  writing  under  seal,  by  which  a 
party  binds  himself  to  pay  the  holder  a  certain  sum,  at  or 
before  a  specified  time. 

405.  What  are  stocks?  An  incorporated  company?  Note.  Into  what  are 
stocks  divided  ?  What  are  stockholders  ?  406.  What  are  certificates  of  stock  ? 
407.  What  is  the  par  value  of  stock?  The  market  value?  Note.  When  are 
stockn  at  par?  Above  par?  Below  par?  The  meaning  of  the  term  par? 
40S.  What  are  the  gross  earnings  of  a  company  ?  The  net  earuiu^'s  ?  409.  Whal 
nre  Instalments ?    410.  Dividends?    411.    What  is  a  bond? 


STOCK    IlirVESTMEKTS.  28:1 

UNITED    STATES    BONDS. 

412.  United  States  Bonds  are  those  issued  by 
Government,  and  are  divided  into  two  classes :  those  pay- 
able at  a  give7i  date,  and  those  payable  within  the  limits 
of  two  given  dates,  at  the  option  of  the  Government. 

Note. — The  former  are  designated  by  a  combination  of  the 
numerals,  which  express  the  rate  of  interest  they  bear,  and  the  year 
they  become  due  ;  as  "  6s  of  '8i." 

The  latter  are  designated  by  combining  the  numerals  expressing 
the  two  dates  between  which  they  are  to  be  paid  ;  as  "  5-203." 

413.  A  Coupon  is  a  certificate  of  interest  attached  to 
a  bond,  which,  on  the  payment  of  the  interest,  is  cut  oif 
and  delivered  to  the  payor. 

414.  The  principal  XJ.  S.  bonds  are  the  following : 

1.  "6s  of  '81,"  bearing  6%  interest,  and  payable  in  1881. 

2.  "  5-20S,"  bearing  dfo  interest,  and  payable  in  not  less  than  5  or 
more  tlian  20  years  from  their  date,  at  the  pleasure  of  the  Govern- 
ment.    Interest  paid  semi-annually  in  gold. 

3.  "  10-40S,'  bearing  5  %  interest,  redeemable  after  10  years  from 
their  date,  interest  semi-annually  in  gold. 

4.  "  5s  of  '81,"  bearing  5%  interest,  redeemable  after  1881,  interest 
paid  quarterly  in  gold. 

5.  "4^s  of  '86,"  bearing  ^\%  interest,  redeemable  after  1886,  the 
interest  paid  quarterly  in  gold. 

6.  "4s  of  1901,"  bearing  4%  interest,  the  principal  payable  after 
1901,  the  interest  paid  quarterly  in  gold. 

415.  State,  City,  I^ailroad  Bonds,  etc.,  are  payable  at  a  specified 
time,  and  are  designated  by  annexing  the  numeral  denoting  the  rate 
of  interest  they  bear,  to  the  name  of  the  State,  etc.,  by  which  they 
are  issued ;  as.  New  York  6s ;  Georgia  7s. 

416.  Computations  in  stocks  and  bonds  are  founded 
upon  the  principles  of  percentage;  the  par  value  being 
the  base,  the  ^;er  cent  premium  the  rate,  the  premium,  etc., 
the  percentage,  and  the  market  value  the  amount. 

412.  What  are  U.  S.  bonds?  How  many  classes?  Note.  How  are  the  former 
designated  ?  The  latter  ?  413.  What  is  a  coupon  ?  414.  What  is  meant  by  TJ.  S. 
^  of  '81  f    By  U.  S.  5-203  ?    By  U.  S.  10-40B  ?    By  new  U.  S.  58  of  '81  ? 


^2  STOCK    IKYESTMENTS 

Note.— Tlie  comparative  profit  of  inveBtments  in  U.  S.  Bonds  de- 
pends upon  their  market  value,  the  rate  of  interest  tliey  bear,  and 
the  premium  on  ^old.  That  of  Railroad  and  other  Stocks  upon  their 
market  value,  and  the  per  cent  of  their  dividends. 

417.  To  find  the  Preiiiium,  Discount,  Tnstahnienf,  or 
Dividend^  the  Par  Value  and  the  Rate  being  given. 

1.  What  is  the  premium  on  20  shares  of  the  New  Orleans 
National  Bank,  at  8^  ?     . 

Analysis. — 2os.=$2ooo;  and  $2000  x  .o8  =  $i6o,  Ans.   Hence,  the 

Rule. — Multij^ily  the  par  value  hy  the  rate,  expressed 
decimally.     (Art.  336.) 

2.  What  is  the  discount  on  27  shares  of  Michigan 
Central  Railroad  at  g%  ? 

3.  What  is  the  premium  on  three  81000  U.  S.  6s  of  ^81, 
when  they  stand  at  17  J;^  above  par  ? 

4.  The  Maryland  Coal  Company  called  for  an  instalment 
of  i$%\  what  did  a  man  pay  who  owned  35  shares? 

5.  What  is  the  discount  on  $4000  Tennessee  6s,  at  10^? 

6.  The  Virginia  Manufacturing  Company  declared  a 
dividend  of  17/^:  to  what  sum  were  28  shares  entitled? 

418.  To  find  thf^  3Iai'kef  Value  of  Stocks  and  Bonds,  the 

Rate  and  Nominal  Value  being  given. 

7.  What  is  the  market  value  of  45  shares  of  New  York 
Central,  at  4%  premium  ? 

Analysis.— 45s.  =  $4500;  and  $4500x1.04  (i  +  the  rate)=$468o, 
the  value  required.     Hence,  the 

Rule. — Multip)ly  the  nominal  value  hy  1  plus  or  minus 
the  rate. 

Or,  multiply  the  market  value  of  i  share  hy  the  number 

cf  shares. 

Note. — In  finding  the  net  value  of  stocks,  the  hrokerage,  postage 
stamps,  and  other  expenses  must  be  deducted  from  the  market  value, 

8.  What  is  the  worth  of  1 14000  of  Kentucky  6s,  at  92  ? 

417.  How  find  the  premium,  etc.,  when  the  par  value  and  rate  are  given  ? 


STOCK    II^VESTMENTS.  283 

9.  What  is  the  worth  of  an  investment  of  $9500  in 
(J.  S.   5-20S,  at   io|^;^  premium  ? 

10.  What  is  the  net  value  of  58  shares  of  New  Jersey 
Central,  at  no;  deducting  the  express  and  other  charges 
$1.89,  and  brokerage  at  i%  ? 

11.  What  will  be  realized  from  $7500  Texas  7s,  at  15^ 
discount,  deducting  the  brokerage  at  ^%,  and  $1.39  for 
postage  and  other  charges. 

12.  What  will  $10000  U.  S.  5-20S  cost  at  S^^o  premium, 
adding  brokerage  at  I'r,  and  other  expenses  $2.37^? 

13.  What  is  the  value  in  currency  of  $15750  gold  coin, 
the  premium  being  2  2j;7^? 

14.  A  person  has  $10000  U.  S.  5-20S:  what  will  he  ra- 
ceive  annually  in  currency,  when  gold  is  12^'  premium? 

419.    To  find   the  Mate,  the   Par  Value,  and  the  Dividend, 
Premium,  or  Discount  being  given. 

15.  The  capital  stock  of  a  company  is  $100000;  its 
gross  earnings  for  the  year  are  $34500,  and  its  expenses 
$13500:  deducting  from  its  net  earnings  $1000  as  surplus, 
what  per  cent  dividend  can  the  company  make  ? 

Analysis. — $13500  +  $1000  =  $14500,  and  $34500  —  $14500  = 
$20000,  the  net  earnings.  Now  20000-7- $100000=. 20  or  20^,  the 
rate  required.     Hence,  the 

EuLE. — Divide  the  premium  or  discount,  as  the  case  may 
be,  hy  the  par  value.     (Art.  339.) 

16.  Paid  $750  premium  for  50  shares  of  bank  stock: 
what  was  the  per  cent  ? 

17.  The  discount  on  100  shares  of  the  Pacific  Eailroad 
is  $625  :  what  is  the  per  cent  below  par? 

18.  The  net  earnings  of  a  company  with  a  capital  of 
I480000  are  $35000;  reserving  $3000  as  surplus,  what 
per  cent  dividend  can  they  declare  ? 

/.18.  How  find  the  market  value  when  the  rate  and  nominal  value  are  given  S 
41).  How  find  the  rate,  when  the  premium,  discount,  or  dividend  are  given? 


284  STOCK    i:N^VESTME:NrTS. 

420.  To  find  how  much  Stock  can  be  bought  for  a  specified 
sum,  the  Rate  of  Premium  or  Discount  being  given. 

19.  How  much  of  Kansas  6s,  at  20^  discount,  can  be 
bought  for  $5200? 

Ai!^'ALYSis. — $5200-^.80  (val.  of  $1  stock)=$65oo,  Ans.    Hence,  the 

Rule. — Divide  the  sum  to  be  invested  hy  i  plus  or  minus 
the  rate,  as  the  case  may  he.     (Art.  349.) 

Or,  divide  the  given  sum  hy  the  marhet  value  of%\  of  stock, 

20.  How  many  $100  U.  S.  10-40S,  at  ^%  premium,  can 
be  bought  for  I4200  ? 

2 1.  What  amount  of  Virginia  6s,  at  90,  can  be  purchased 
for  1 1 0800? 

421.  To  find  what  Sum  must  be  invested  in  Bonds  to  realize 
a  given  income,  the  Cost  and  Rate  of  Interest  being  given. 

22.  What  sum  must  be  invested  in  Missouri  6s  at  90,  to 
reahze  an  income  of  %  1 800  annually  ? 

Analysis. — At  6%  the  income  of  $1  is  6  cts. ;  and  $1800-4- .06= 
$30000,  the  nominal  value  of  the  bonds.  Again,  at  .90,  $30000  of  bonds 
will  cost  .90  times  $30000= $27000,  the  sum  required.     Hence,  the 

Rule. — I.  Divide  the  given  income  hy  the  annual  interest 
9/  1 1  of  honds;  the  quotient  ivill  he  the  jiominal  value  of 
the  honds. 

II.  Multiply  the  nominal  value  hy  the  marhet  value  of$i 
of  honds  J  the  product  ivill  he  the  sum  required. 

23.  What  sum  must  be  invested  in  IJ.  S.  5-20S,  at  106, 
to  yield  an  annual  income  of  ^2500  in  gold  ? 

24.  How  much  must  one  invest  in  Wisconsin  8s,  at  95, 
to  receive  an  annual  income  of  $3000  ?  Ans.  $35625. 

25.  How  much  must  be  invested  in  Mississippi  6s,  at  80, 
to  yield  an  annual  interest  of  $4200  ? 

420.  How  find  the  quantity  of  stock  that  can  be  bought  for  a  epecifled  sum, 
when  the  rate  is  given  ? 


EXCHANGE. 

422.  JExchange  is  a  method  of  making  payments 
between  distant  places  by  Bills  of  Exchange. 

423.  A  Bill  of  Exchange  is  an  order  or  draft 
directing  one  person  to  pay  another  a  certain  sum  at  a 
specified  time. 

Note. — The  person  who  signs  the  bill  is  called  the  drawer  or 
maker ;  the  one  to  whom  it  is  addressed,  the  drawee;  the  one  to 
whom  it  is  to  be  paid,  the  payee  ;  t*he  one  who  sends  it,  the  remitter. 

424.  Bills  of  Exchange  are  Domestic  or  Foreign. 
Domestic  Bills  are  those  payable  in  the  country 

where  they  are  drawn,  and  are  commonly  called  Drafts. 

Foreign  Bills  are  those  drawn  in  one  country  and 
payable  in  another. 

425.  A  Sight  Bill  is  one  payable  on  its  presentation, 

A  Time  Bill  is  one  payable  at  a  specified  time  after 
its  date,  or  presentation. 

426.  The  Bar  of  Exchange  is  the  standard  by 
which  the  value  of  the  currency  of  difierent  countries  is 
compared,  and  is  either  i^itrinsic  or  no?mnal. 

An  Intrinsic  Bar  is  a  standard  having  a  real  and 
fixed  value  represented  by  gold  or  silver  coin. 

A  N^oininal  Bar  is  a  conventional  standard,  having 
any  assumed  value  which  convenience  may  suggest. 

427.  When  the  market  price  of  bills  is  the  same  as  the 
face,  they  are  at  par ;  when  it  exceeds  the  face,  they  are 
above  par,  or  at  a  premium;  when  it  is  less  than  the  face, 
they  are  helow  par,  or  at  a  discount.     (Art.  407,  n.) 

Notes.— I.  The  fluctuation  in  the  price  of  bills  from  their  par 
f*lue,  is  called  the  Course  of  Exchange. 

42a.  What  is  exchange  T  423.  A  bill  of  exchange  ?  424.  What  are  Jomestic 
bflla  T  Foreign  bills  ?  425.  What  are  eight  bills  ?  Time  bills  ?  436.  What  ia 
the  par  of  exchange  ?    An  intrinsic  par  ?    A  nominal  par  ? 


28G  DOMESTIC     EXCHANGE. 

2.  The  rate  of  Exchange  between  two  places  or  countries  depen^'a 
upon  the  circumstances  of  trade.  If  the  trade  between  New  York 
and  New  Orleans  is  equal,  exchange  is  at  par.  If  the  former  owls 
the  latter,  the  demand  for  drafts  on  New  Orleans  is  greater  than  the 
supply;  hence  they  are  above  par  in  New  York.  If  the  latter  owes 
the  former,  the  demand  for  drafts  is  less  than  the  supply,  con- 
sequently drafts  on  New  Orleans  are  helou)  par. 

428.  An  accex>tance  of  a  bill  or  draffc  is  an  engage- 
ment  to  pay  it  according  to  its  conditions.  To  show  this, 
it  is  customary  for  the  drawee  to  write  the  word  accepted 
across  the  face  of  the  bill,  with  the  date  and  his  name. 

Note. — Bills  of  Exchange  are  negotiable  like  promissory  notes, 
and  the  laws  respecting  their  indorsement,  collection,  protest,  etc., 
are  essentially  the  same. 


DOMESTIC    OR    INLAND    EXCHANGE. 

429.  To  find  the  Cost  of  a    Draft,  its   Face  and  the  Ra'.e 
of  Exchange  being  given. 

1.  What  is  the  cost  of  a  sight  draft  on  New  York,  for 
I4500,  at  2\%  premium  ? 

Analysis.— At  2\%  premium,  a  draft  of  |i  will  I4500 

cost  $1  +  2^  cts.  =  $i.o25,  or  1.025  times  the  draft.  I.025 

Hence,  the  cost  of  $4500  draft  will  be  1.025  times  a        ^TtTs^co 
its  face  ;  and  $4500  X  i.o25=$46i2. 50,  ^?i«.  '^ 

2.  "VYhat  is  the  cost  of  a  sight  draft  on  St.  Louis  of 
$5740,  at  4%  discount  ? 

Analysis. — At  4^  discount,  a  draft  of  $1  will  $5740 

cost   $1—4  cents=$0  96,  or  .96  times  the  draft  .q6 

Therefore,  the  cost  of  $5740  draft  will  be  .06  times  i7T7r~rr~  ^  ^,  „ 

ats  face ;  and  $5740  x  .96=15510.40,    Hence,  the  ^^ 

EuLE. — Multiply  the  face  of  the  draft  ly  the  cost  of  $1 
of  draft,  expressed  decimally. 

427.  Whan  are  bills  at  par  ?  When  ahove  par  ?  When  below  ?  Note.  What 
is  the  fluctuation  in  the  price  of  exchange  called  ?  Upon  what  does  the  rate  of 
excuange  depend  ?  Explain?  428.  The  acceptance  of  a  bill?  429.  How  find  the 
'JOBt  of  a  draft,  when  the  face  and  rate  are  given  ? 


DOMESTIC     EXCHA2^aE.  287 

Notes. — i.  When  payable  at  sight,  the  worth  of  $i  of  draft  is  $i 
p?MS  or  minus  the  rate  of  exchange. 

2.  On  time  drafts,  both  the  exchange  and  the  bank  discount  are 
computed  on  their  face.  Dealers  in  excliange,  however,  make  but. 
one  computation  ;  the  rate  for  time  drafts  being  enough  less  than 
sight  drafts  to  allow  for  the  bank  discount. 

2.  A  merchant  in  Galveston  bouglit  a  draft  of  $2000  on 
Philadelphia  at  60  days  sight :  what  was  the  draft  worth, 
the  premium  being  3^^,  and  the  bank  discount  Gfc 

3.  What  cost  a  sight  draft  of  $3560,  at  2^  discount? 
^  4.  Eequired  the  worth  of  a  draft  for  84250  on  Chicago, 
at  90  days,  sight  drafts  being  1%  discount  and  interest  "]%. 
^  5.  The  Bank  of  New  York  having  declared  a  dividend 

of  4^,  a  stockholder  living  in  Savannah  drew  on  the  bank 
for  his  dividend  on  50  shares,  and  sold  the  draft  at  \\% 
premium :  how  much  did  he  realize  from  the  dividend  ? 

430.  To  find  the  Face  of  a   Draft,  the  Cost  and   Rate  of 
Exchange  being  given. 

7.  Bought  a  draft  in  Omaha  on  New  York,  payable  in 
90  days,  for  $3043.50,  exchange  being  2>%  premium,  and 
interest  (i% :  what  w^as  the  face  of  the  draft  ? 

Analysis. — The  cost  of  $r  of  |j.  ^_  ^a/__^j_q^ 

draft  is  $1  plus  the  rate,  minus     pjg^  ^^  ^^  fo^,  ^^^^^o  oj^^ 

the  bank  discount  for  the  time.  .-.     i     r.  a     ■\dl      is 

Now  $i  +  3%-|i.o3;    the   bank  Cost  of  $i  dft.  =  $l.oi45 

discount  on  $1  for  93  d.  is  $0.0155,  i-OM5)^3043-5o 

and    $1.03  -  $0.0155  =  $10145.        Face  of  dft.=: $3000. 
Now  if  I1.0145  will  buy  $1   of 

draft,  $3043.50  will  buy  a  draft  of  as  many  dollars  as  1.0145  is  con- 
tained times  in  $3043.50.  $3043.50-^1.0145  — $3000,  the  draft  re- 
quired.   Hence,  the 

KuLE. — Divide  the  cost  of  the  draft  ly  the  cost  of  %i 
of  draft,  expressed  decimally. 

8.  What  is  the  face  of  a  sight  draft  purchased  for  $1250, 
the  premium  being  2^%  ? 

430.  How  find  the  face  of  a  draft,  the  cost  and  rate  being  given  ? 


288  ¥OREIGKEXCHAKGE. 

9.  What  is  the  face  of  a  draft  at  60  days  sight,  purchased 
for  1 1 500,  when  interest  is  8^,  and  the  premium  2%? 
^A   -^10.  What  is  the  face  of  a  sight   draft,  purchased   for 

$2500,  the  discount  being  4^^  ? 
y/  —  1 1.  What  is  the  face  of  a  draft  on  4  m.,  bought  for  I3600, 
^      the  int.  being  6^,  and  exchange  2%  discount  ? 
J"^     12.  A  merchant  of  Natchez  sold  a  draft  on  Boston  at 
*^        r^%  prem.,  for  $3806.25  :  what  was  the  face  of  the  draft  ? 

FOREIGN     EXCHANGE, 

431.  A  Foreign  Bill  of  Uxchanr/e  is  a  Bill 
drawn  in  one  country  and  payable  in  another. 

A  Set  of  Exchange  consists  of  three  bills  of  the 
same  date  and  tenor,  distinguished  as  the  First,  Second, 
and  Third  of  exchange.  They  are  sent  by  different  mails, 
in  order  to  save  time  in  ease  of  miscarriage.  When  one  is 
paid,  the  others  are  void. 

432.  The  Legal  JPar  of  Exchange  between 
Great  Britain  and  the  United  States,  is  $4.8665  gold  to  the 
pound  sterling.* 

433.  To  find  the  Cost  of  a   Bill   on   England,  the    Face   and 
the  Rate  of  Exchange  being  given^ 

I.  What  is  the  cost  of  the  following  bill  on  London,  at 
$4.8665  to  the  £  sterhng  ? 

^354>  1 28-  New  Yobs,  July  4th,  1874. 

At  sight  of  this  first  of  exchange  (the  second  and  third 
of  the  same  date  and  tenor  unpaid),  pay  to  the  order  of 
Henry  Crosby,  three  hundred  and  fifty-four  pounds,  twelve 
shillings  sterling,  value  received,  and  charge  the  same  to 
the  account  of  0.  J.  King  &  Co. 

To  George  Peabodt,  Esq.,  London. 

431.  What  is  a  foreign  bill  of  exchange  ?  What  is  the  legal  par  of  exchang* 
with  Great  Britain  ? 

*  Act  of  Congress,  March  3d,  1873— To  t*^^  eflbct  Jan.  ist,  1874. 


FOREIGN     EXCHAI^GE.  289 

Analysis. — Reducing  tlie  given  sliillings  to  opbbation. 

the  decimal  of  a  pound,  £354,  12s.  —  £354.6.  4.8665 

(Art.  295.)    Now  if  £1  is  worth  $4.8665,  £354-6  3_54'^ 

are  worth  354.6  times  as  much;  and  $4.8665  x  Ans,  $1725.661 
354.6=$i725.66i,  the  cost  of  the  bill. 

2.  What  is  the  value  of  a  bill  on  England  for  ^£43 6,  5  s.  6d., 
at  $4.85  J  to  the  £  sterling  ? 

Analysis. — £436,  5s.  6d.=£436.275,  and  the  operation. 

market  value  of  exchange  is  $4.8525  to  the  £.  $4.8525 

Now $4.8525  X436.275=$2ii7.02,  the  cost  of  the  436.275 

bill.    Hence,  the  A^g.  "$2117.02 

Rule. — Multiply  the  market  value  of  £1  sterling  ly  the 
face  of  the  bill ;  the  product  will  be  its  value  in  dollars 
and  cents. 

Notes. — i.  If  there  are  shillings  and  pence  in  the  given  bill, 
they  should  be  reduced  to  the  decimal  of  a  pound.    (Art.  295.) 

2.  Bills  on  Great  Britain  are  drawn  in  Sterling  money. 

3.  The  New  Par  of  Sterling  Exchange  $4.8665,  is  the  intrinsic 
value  of  the  Sovereign  or  pound  sterling,  as  estimated  at  the  United 
States  Mint,  and  is  9?  %  greater  than  the  old  par. 

4.  The  Old  Par,  which  assumed  the  value  of  the  £  sterling  to  be 
$4.44^,  is  abolished  by  law,  and  all  contracts  based  upon  it  after 
January  ist,  1874,  are  nvM  and  void. 

3.  What  is  the  cost  of  a  bill  on  DubUn  for  ^£381,  at 
$4.87!-  to  the  ie  sterling  ? 

4.  What  is  the  cost  in  currency  of  a  bill  on  England  for 
^6750,  exchange  being  at  par,  and  gold  2>^l%  premium? 

5.  B  owes  a  merchant  in  Liverpool  ^£1500;  exchange  is 
$4.93!;  to  transmit  coin  will  cost  2^  insurance  and  freight; 
and  when  delivered  its  commercial  value  is  $4.80  to  a 
} sound:  which  is  the  cheaper,  to  buy  a  bill,  or  send  the 
gold  ?    How  much  ? 

6.  A  New  Orleans  merchant  consigned  568  bales  of  cot- 
^  ton,  weighing  450  lbs.  apiece,  to  his  agent  in  Liverpool; 

the  agent  paid  id.  a  pound  freight,  and  sold  it  at  i2d.  a 

433.  How  find  the  cost  of  a  bill  on  England,  when  the  face  and  rate  are  given  ? 


290  FOREIG^q"     EXCHANGE. 

pound ;  he  charged  2  J^  commission,  and  £S,  6s.  for  stor- 
age: what  did  the  merchant  realize  for  his  cotton,  ex- 
change on  the  net  proceeds  being  1491^  to  the  £ ? 

434.    To  find  Ihe  Face  of  a  Bill  on  England,  the  Cost  and 
the  Rate  of  Exchange  being  given* 

7.  What  is  the  face  of  a  bill  on  England  which  cost 
$1725.72,  exchange  being  at  par,  or  I4.8665  to  the  £? 

ANALYsis.-Smce  $4.8665  buys  £1  of  $4.866s)li72t;.72 

bill,  $1725.72  will  buy  as  many  pounds  as  — ^— — — — 

$4.8665  are  times  in  $1725.72,  or  £354.612  *'354-oi2 

=£354,  I2S.  2|d.    Hence,  the  ^^^5.  £354,  12s.  2jd. 

KuLE. — Divide  the  cost  of  the  hill  ty  the  market  value 
of  £1  sterling  ;  the  quotient  will  he  the  face  of  the  hill. 

8.  What  is  the  face  of  a  bill  on  London  which  cost 
$2500,  exchange  being  $4.88|  to  the  £'^ 

9.  What  amount  of  exchange  on  Dubhn  can  be  obtained 
for  $3750,  at  $4.84^  to  the  £  sterling? 

10.  A  merchant  in  Charleston  paid  $5000  for  a  bill  on 
London,  at  $4.87^  to  the  £-.  what  was  the  face  of  the  bill? 

11.  Paid  1 7 500  for  a  bill  on  Manchester:  what  was  the 
face  of  it,  exchange  being  I4.86J-  to  the  iB? 

434,  a.  In  quoting  Exchange  on  Foreign  Countries,  the  gen- 
eral rule  is  to  quote  the  money  of  one  country  against  the  money 
of  other  countries.     Thus,  exchange  is  quoted 

On  Austria,  in  cents  to  the  Florin  (silver) =$0476. 

On  Frankfort,  in  cents  to  the  Florin  (gold) =$0.4165. 

On  the  German  Empire,  in  cents  to  the  Mark  (gold)= $0.2382. 

On  North  Germany,  in  cents  to  the  new  Thaler  (silver) =$0.7 14. 

On  Russia,  in  cents  to  the  Rouble  (silver) =$0.7717. 

Note. — Bills  of  Exchange  between  the  United  States  and  foreign 
countries  are  generally  drawn  on  some  of  the  great  commercial 
centers ;  as  London,  Paris,  Frankfort,  Amsterdam,  etc. 

434.  IIow  find  the  face  of  a  bill  on  Bngland,  when  the  cost  and  rate  are  given  ? 


FOREIGN"     EXCHANGE.  291 

435.  Bills  Oil  France  are  drawn  in  French  cur- 
rency, and  are  calculated  at  so  manj  francs  and  centimes 
to  a  dollar. 

Note. — Centimes  are  commonly  written  as  decimals  of  a  franc 
Tims  5  francs  and  23  centimes  are  written  5.23  francs.     (Art.  224,  n.) 

436.  To  find  the   Cost  of  a   Bill   on   France,  the   Face  ana 

Rate  of  Exchange  being  given. 

12.  What  is  the  cost  of  a  bill  on  Paris  of  1500  francs, 
exchange  being  5.25 f  to  a  dollar? 

Analysis. — Since  5.25  francs  will  buy  a  5.25  fr.)  1500.00  fr. 
bill  of  $1,  1500  francs  will   buy  a  bill  of  as  Ans  "^28=;  7^  4- 

many  dollars  as  5.2',  is  contained   times  in 
1500;  and  1500-7-5. 25  =  $285. 71,  the  cost  required.     Hence,  the 

Rule. — Divide  the  face  of  the  Mil  ly  the  number  of  francs 
to  %i  exchange. 

13.  What  is  the  cost  of  a  bill  of  3500  francs  on  Havre, 
exchange  being  5.18  francs  to  a  dollar? 

437.  To  find  the  Face  of  a  Bill   on  France,  the  Cost  and 

the  Rate  of  Exchange  being  given. 

14.  A  traveler  paid  I300  for  a  bill  on  Paris;  exchange 
being  5.16  francs  to  $1 :  what  was  the  face  of  the  bill  ? 

Analysis. — If  $1  will  buy  5.16  francs,  $300  will  buy  300  times  as 
many  ;  and  5.16  fr.  x  300=1548  francs,  the  face  of  the  bill  required. 
Hence,  the 

Rule. — Multij^ly  the  number  of  francs  to  %i  exchange  bt, 
the  cost  of  the  bill. 

15.  Paid  $2500  for  a  bill  on  Lyons,  exchange  being 
5.22  francs  to  $1 :  what  was  the  face  of  the  bill  ?  • 

16.  A  merchant  paid  $3150  for  a  bill  on  Paris,  exchange 
being  5.23  francs  to  |i :  what  was  the  face  of  the  bill  ? 

435.  How  is  exchangee  on  France  calculated  ?  436.  How  find  the  cost  of  a  bill 
on  France,  when  the  face  and  rate  are  given?  437.  How  find  the  face,  when  the 
cost  and  rate  are  gireu  ? 


438.  Insurance  is  indemnity  for  loss.  It  is  dis- 
tinguished by  different  names,  according  to  the  cause  of 
the  loss,  or  the  object  insured. 

439.  Fif^e  Insurance,  is  indemnity  for  loss  by  fire. 
Marine  Insurance,  for  loss  by  sea. 

Life  Insurance,  for  the  loss  of  life. 
A.ccident  Insurance,  for  personal  casualties. 
Health  Insurance,  for  personal  sickness. 
Stocic  Insurance,  for  the  loss  of  cattle,  horses,  etc. 

Notes. — i.  The  party  who  undertakes  the  risk  is  called  the 
Insurer  or  Underwriter. 

2.  The  party  protected  by  the  insurance  is  called  the  Insured. 

440.  A  JPolicy  is  a  writing  containing  the  evidence 
and  terms  of  insurance. 

The  Premiiini  is  the  sum  paid  for  insurance. 

Note. — The  business  of  insurance  is  carried  on  chiefly  by  In- 
corporated  Companies.  Sometimes,  however,  it  is  undertaken  by 
individuals,  and  is  then  called  out-door-insurance. 

441.  Insurance  Companies  are  of  two  kinds :  Stock  Companies, 
and  Mutual  Companies. 

A  Stock  Insurance  Company  is  one  which  has  a  paid  up 
capital,  and  divides  the  profit  and  loss  among  its  stockholders. 

A  Mutual  Insurance  OoJW^atii/ is  one  in  which  the  lossea 
are  shared  by  the  parties  insured. 

442.  Premiums  are  computed  at  a  certain  per  cent  of 
the  sum  insured,  and  the  operations  are  similar  to  those 
in  Percentage. 

Note. — Policies  are  renewed  annually,  or  at  stated  periods,  and 
the  premium  is  paid  in  advance.  In  this  respect  insurance  diflFera 
from  commission,  etc.,  which  have  no  reference  to  time. 

438.  What  is  insurance?  439.  Fire  insurance?  Marine?  Life?  Accident? 
Health?  Stock?  440.  What  is  a  policy  ?  The  premium?  441.  How  many  kinds 
of  insurance  companies?  A  stuck  insurance  company?  A  mutual?  442.  How 
are  premiums  computed  ? 


II^SURAKCE.  293 

FIRE     AND     MARINE     INSURANCE. 

443.  To  find  the  Preniui^n,f  the  Sum  insured  and  the  Rate 

for  the  period  being  given. 

1.  What  premium  must  I  pay  per  annum  for  insuring 
^1750  on  my  house  and  furniture,  at  ^%'t 

Analysis. — $1750  x  .005  (rate)=$8.75,  Ans.    Hence,  tlie 

EuLE. — Multiply  the  sum  insured  ly  the  rate.   (Art.  2,2)^.) 

Note. — The  rate  of  insurance  is  sometimes  stated  at  a  certain 

number  of  cents  on  $100.     In  such  cases  tlie  rate  should  be  reduced 

io  the  decimal  of  $1  before  multiplying.    Thus,  if  the  rate  is  25  cents 

on  $100,  the  multiplier  is  written  .0025.     (Art.  295.) 

2.  At  35  cents  on  $100,  what  is  the  insurance  of  $1900? 

3.  At  i\%,  what  is  the  premium  on  $2560? 

4.  At  2^^,  what  is  the  premium  on  $3750? 

5.  At  25  cents  on  $100,  what  is  the  premium  on  $4280? 

6.  At  50  cents  on  $100,  what  is  the  premium  on  $5000? 

7.  At  3^^,  what  is  the  cost  of  insuring  $6175  ? 

8.  What  is  the  premium  for  insuring  a  ship  and  cargo 
valued  at  $35000,  at  2Y/c  ? 

9.  What  is  the  cost  of  insuring  a  factory  and  its  con- 
tents, valued  at  I48250,  at  3 J;|,  including  %i\  for  policy  ? 

444.  To  find  what  Sum  must  be  insured  to  cover  both  the 

Property  and  Premium,  the  Rate  being  grven. 

10.  For  what  must  a  factory  worth  $20709  be  insured, 
to  cover  the  property  and  the  premium  of  2^%  ? 

Analysis. — The  sum  to  be  insured  includes  the  property  plus  the 
premium.  But  the  property  is  100%  of  itself,  and  the  premium  is 
2.\fc  of  that  sum  ;  therefore  $20709=100%— 2^%,  or  .97^  times  the 
sum  to  be  insured.  Now,  if  $20709  is  .97^  times  the  required  sum, 
once  that  sum  is  $20709 -=-.971,  or  $21240,  Ans.   (Art.  337.)   Hence,  the 

EuLE. — Divide  the  value  of  the  property  by  1  minus  the  rate. 

Note. — This  and  the  preceding  problem  cover  the  ordinary  cases 
of  Fire  and  Marine  Insurance.  Should  other  problems  occur,  they 
may  be  solved  like  the  corresponding  problems  in  Percentage 

How  find  the  premium,  when  the  sum  insured  and  ttie  rate  arc  given  I 


294  I2^SURAKCE. 

11.  A  merchant  sent  a  cargo  of  goods  worth  $15275  to 
Canton:  what  sum  must  he  get  insured  at  2>%,  that  he 
may  suffer  no  loss,  if  the  ship  is  wrecked  ? 

12.  A  house  and  furniture  are  worth  $27250 :  what  sum 
must  be  insured  at  2%  to  cover  ihQ  property  and  premium  ? 

13.  What  sum  must  be  insured  at  <^%  to  coyer  the 
premium,  with  a  vessel  and  cargo  worth  I35250? 

LIFE     INSURANCE. 

445.  Life  Insurance  Policies  are  of  different  hinds,  and 
the  premium  varies  according  to  the  expectation  of  life. 

ist.  Life  Policies^  whicli  are  payable  at  the  death  of  the  party- 
named  in  the  policy,  the  annual  premium  continuing  through  life. 

2d.  Lfife  l^oUcieSf  payable  at  the  death  of  the  insured,  the 
annual  premium  ceasing  at  a  given  age, 

3d.  Term  Policies ,  payable  at  the  death  of  the  insured,  if  he 
dies  during  a  given  term  of  years,  the  annual  premium  continuing 
till  the  policy  expires. 

4th.  JEhidotvinent  Policies,  payable  to  the  insured  at  a  given 
age,  or  to  his  heirs  if  he  dies  before  that  age,  the  annual  premium 
continuing  till  the  policy  expires. 

Note. — The  expectation  of  life  is  the  average  duration  of  the  life 
of  individuals  after  any  specified  age. 

1.  What  premium  must  a  man,  at  the  age  of  25,  pay 
annually  for  a  life  policy  of  $5000,  at  4^%  ? 

Analysis. — 4^%  =,045,  and  $5000  x  .045  =  $22 5,  Ans.     (Art.  443.) 

2.  What  is  the  annual  premium  for  a  life  policy  of 
$2500,  at  5^? 

3.  A  man  at  the  age  of  35  years  effected  a  life  insurance 
of  I7500  for  10  yrs.,  at  3^^:  what  was  the  amt.  of  premium? 

4.  A  man  65  years  old  negotiated  a  life  insurance  of 
$8000  for  5  years,  at  1 2}^ :  what  was  the  amt.  of  premium  ? 

5.  At  the  age  of  30  years,  a  man  got  his  life  insured  for 
^75000,  at  4%  per  annum,  and  lived  to  the  age  of  70 :  which 
was  the  greater,  the  sum  he  paid,  or  the  sum  received? 

444.  How  find  what  sum  must  be  insured  to  cover  the  property  and  premium  1 


TAXES. 

446.  A  Tax  is  a  sum  assessed  upon  the  person  or 
property  of  a  citizen,  for  public  purposes. 

A  Property  Tax  is  one  assessed  upon  jjroperty. 

A  JPersonal  Tax  is  one  assessed  upon  the  perso?i, 
and  is  often  called  a  poll  or  capitatioji  tax. 

Note. — The  term  poll  is  from  tlie  German  polU,  tlie  liead ;  capita- 
tion, from  the  Latin  caput,  the  head. 

447.  I*roperty  is  of  two  kinds,  real  and  personal. 
Ileal  T*roperty  is  that  which  is  fixed;  as,  lands, 

houses,  etc.     It  is  often  called  real  estate. 

JPersonal  Property  is  that  which  is  movable;  as, 
money,  stocks,  etc. 

448.  An  Assessor  is  a  person  appointed  to  appraise 
property,  for  the  purpose  of  taxation. 

449.  An  Inventory  is  a  list  of  taxable  property, 
with  its  estimated  value. 

450.  Pfoperty  taxes  are  computed  at  a  certain  per  cent  on  the 
valuation  of  the  property  to  be  assessed.     Tiiat  is. 

The  valuation  of  the  property  is  the  base ;  the  sum  to  be  raised, 
the  percentage ;  the  per  cent  or  tax  on  $i,  the  rate  ;  the  sum  collected 
minus  the  commission,  the  net  proceeds. 

Poll  taxes  are  specific  sums  upon  those  not  exempt  by  law,  without 
regard  to  property. 

451.    To  assess  a  Property  Tax,  the  Valuation  and  the 
Sum  to  be  raised  being  given. 

I.  A  tax  of  $6250  was  levied  upon  a  corporation  of  6  per- 
sons; A's  property  was  appraised  at  $30000;  B's,  I37850; 
C's,  $40150 ;  D's,  $50000 ;  E's,  $55000 ;  Fs,  $37000.  What 
was  the  rate  of  the  tax,  and  what  each  man's  share  ? 

446.  What  is  a  tax  ?  A  property  tax  ?  A  personal  tax  ?  Note.  What  is  a  per- 
sonal tax  called  ?  447.  Of  how  many  kinds  is  property  ?  What  is  real  property  ? 
Personal  ?  448.  What  is  an  assessor  ?  449.  An  inventory  ?  450.  How  are  prop- 
erty taxea  computed?  How  are  poll  taxes  levied?  451.  How  is  a  property  tax 
assessed,  when  the  valuation  and  the  sum  to  be  raised  are  given  ? 


296 


TAXES. 


AnsALYsis. — The  valuation=$300oo  + $37850+ $40150+ $500004- 
$55000  + $37000=  $250000,  The  valuation  $250000,  is  the  hase ; 
and  the  tax  $6250,  the  percentage.  Therefore  $625o-t-$25oooo=.025, 
or  2hfc,  the  rate  required. 

Again,  since  A's  valuation  was  $30000,  and  the  rate  2^5^,  his  tax 
must  have  been  2i%  of  $30000;  and  $30000  x  .025 =$750.00.  The 
tax  of  the  others  may  be  found  in  like  manner.     Hence,  the 

EuLE. — I.  MaJce  an  inventory  of  all  the  taxable  property. 

II.  Divide  the  sum  to  le  raised  ly  the  amount  of  the 
inventory y  and  the  quotient  will  he  the  rate. 

III.  Multiply  the  valuation  of  each  man^s  property  hy 
the  rate,  and  the  product  will  le  his  tax. 

Notes. — i.  If  a  poll  tax  is  included,  the  sum  arising  from  the 
polls  must  be  subtracted  from  the  sum  to  be  raised,  before  it  is 
divided  by  the  inventory. 

2.  If  the  tax  is  assessed  on  a  large  number  of  individuals,  the 
operation  will  be  shortened  by  first  finding  the  tax  on  $1,  $2,  $3,  etc., 
to  $9 ;  then  on  $10,  $20,  etc.,  to  $90 ;  then  on  $100,  $200,  etc.,  to  $900, 
etc.,  arranging  the  results  as  in  the  following 

ASSESSORS'    TABLE. 


$1 

pays 

$.025 

$10 

pay 

$.25 

$100 

pay 

$2.50 

2 

a 

.050 

20 

.50 

200 

5.00 

3 

a 

.075 

30 

ii 

•75 

300 

ii 

7-50 

4 

a 

.100 

40 

ii 

1. 00 

400 

.10.00 

5 

a 

•125 

50 

ii 

1-25 

500 

ii 

12.50 

6 

a 

.150 

60 

ii 

1.50 

1000 

ii 

25.00 

7 

a 

•175 

70 

ii 

1-75 

2000 

ii 

50.00 

8 

a 

.200 

80 

ii 

2.00 

3000 

ii 

75.00 

9 

i( 

.225 

90 

ii 

2.25 

4000 

a 

100.00 

2.  B's  yaluation=:$3785o=$3oooo  +  $7ooo  +  $8oo  +  $5o, 
By  the  table  the  tax  on  $30000 =$7  5  0.00 
"  "  7000=   175.00 

"  "  800=:     20.00 

50= 1^ 

Therefore,  we  have  B's  tax,        =$946.25 


Note.  If  a  poll  tax  is  included,  how  proceed  ? 


TAXES.  297 

3.  Required  C,  D,  E,  and  F's  taxes,  both  by  the  rule 
and  the  taUe. 

4.  A  tax  levied  on  a  certain  township  was  $16020;  the 
valuation  of  its  taxable  property  was  $784750,  and  the 
number  of  polls  assessed  at  $1.25  was  260.  What  was  the 
rate  of  tax ;  and  what  was  A's  tax,  who  paid  for  3  polls, 
the  valuation  of  his  property  being  I7800  ? 

5.  The  State  levied  a  tax  of  $165945  upon  a  certain  city 
which  contained  1260  polls  assessed  75  cents  each,  an  in- 
ventory of  I5427600  real,  and  $72400  personal  property. 
What  was  G's  tax,  whose  property  was  assessed  at  $15000  ? 

6.  What  was  H's  tax,  whose  inventory  was  $10250,  and 
3  polls  ? 

7.  A  district  school-house  cost  $2500,  and  the  valuation 
of  the  property  of  the  district  is  $50000  :  what  is  the  rate, 
and  what  A's  tax,  whose  property  is  valued  at  $3400  ? 

452.    To  find  the  Antoiuit  to   be   assessed,  to  raise  a  net 
sum,  and  pay  the  Commission  for  collecting  it. 

8.  A  certain  city  required  $47500  to  pay  expenses:  what 
amount  must  be  assessed  in  order  to  cover  the  expenses, 
and  the  commission  of  5^  for  collecting  the  tax  ? 

Analysis. — At  5  %  for  collection,  $1  assessment  yields  $.95  ; 
tlicrefore,  to  obtain  $47500  net,  requires  as  many  dollars  assessment 
as  $.95  are  contained  times  in  $47500;  and  $47500-?-. 95  (1  —  5^)  = 
$50000,  the  sum  required  to  be  assessed.     (Art.  341.)    Hence,  the 

KuLE. — Divide  the  net  sum  ly  i  miiius  the  rate;  the 
quotient  will  be  the  amount  to  be  assessed.     (Art.  348.) 

9.  What  sum  must  be  assessed  to  raise  a  net  amount  of 
$3500,  and  pay  the  commission  for  collecting,  at  4%  ? 

10.  What  sum  must  be  assessed  to  raise  a  net  sum  of 
$5260,  and  pay  for  the  collection,  at  4^%  commission  ? 

11.  What  sum  must  be  assessed  to  raise  a  net  amount 
of  $10500,  and  pay  the  commission  for  collecting,  at  5^? 

452.  How  find  the  amount  to  be  assessed,  to  cover  the  sum  to  be  raised  and 
the  commission  for  collection  ? 


DUTIES. 

453.  Duties  are  sums  paid  on  imported  goods,  and 
are  often  called  customs. 

454.  A  Custom  House  is  a  building  where  duties  are  re- 
ceived, ships  entered,  cleared,  etc. 

455.  A  Port  of  Entry  is  one  where  there  is  a  Custom  House. 
The  Collector  of  a  Fort  is  an  officer  who  receives  the  duties, 

has  the  charge  of  the  Custom  House,  etc. 

456.  A  Tariff  is  a  list  of  articles  subject  to  duty,  stating  the 
rate,  or  the  sum  to  be  collected  on  each. 

457.  An  Invoice  is  a  list  of  merchandise,  with  the  cost  of  the 
SGveral  articles  in  the  country  from  which  they  are  imported. 

458.  Duties  ^re  of  two  kinds,  specitic  and  ad  yalorem. 

459.  A  Specific  Duty  is  a  fixed  sum  imposed  on 
each  article,  ton,  yard,  etc.,  without  regard  to  its  cost. 

An  Ad  Valorem  Duty  is  a  certain  joer  cent  on  ilie 
value  of  goods  in  the  country  from  which  they  are 
imported. 

Note. — The  term  ad  valorem  is  from  the  Latin  ad  and  valorem, 
according  to  value. 

460.  Before  calculating  duties,  certain  allowances  are  made, 
called  tare,  tret  or  draft,  leakage,  and  breakage. 

Tare  is  an  allowance  for  the  weight  of  the  box,  bag,  cask,  etc., 
containing  the  goods. 

Tret  is  an  allowance  in  the  weight  or  measure  of  goods  for  wa£te 
or  refuse  matter,  and  is  often  called  draft. 

Leakage  is  an  allowance  on  liquors  in  casks. 

JBreaJcage  is  an  allowance  on  liquors  in  bottles. 

Notes. — i.  Tare  is  calculated  either  at  the  rate  specified  in  the 
invoice,  or  at  rates  established  by  Act  of  Congress. 

2.  Leakage  is  commonly  determined  by  gauging  the  casks,  and 
Breakage  by  counting. 

3.  In  making  these  allowances,  if  the  fraction  is  less  than  i  it  is 
rejected,  if  ^  or  more,  i  is  added. 

45^.  What  are  duties?  454.  A  custom  house?  455.  What  is  a  port  of  entry? 
A  collector?  456.  A  tariff?  457.  What  is  an  invoice?  45S.  Of  how  many 
kinds  are  duties  ?  459.  A  specific  duty  ?  An  ad  valorem  ?  460.  What  is  tare  ? 
Tret  ?    Leakage  ?    Breaka<?e  ? 


DUTIES.  299 

P  R  O  B  L  E  M    I . 

461.  To  calculate  Specific  I>vties,  the  quantity  of  goods, 
and  the  Sum  levied  on  each  article  being  given. 

1.  What  is  the  specific  duty  on  12  casks  of  brandy,  each 
containing  40  gal.,  at  $1 J  per  gal.,  allowing  2%  for  leakage  ? 

Analysis. — 12  casks  of  40  gallons  each,  =480.0  gal. 

The  leakage  at  2fo  on  480  gallons,  ==      9.6  gal. 

The  remainder,  or  quantity  taxed,  =470.4  gal. 

Now  $1.50  X  4704  =  $705.6o,  the  duty  required.  Hence,  the 

Rule. — Deduct  the  legal  alloivance  for  tare,  tret,  etc., 
from  the  goods,  and  muUij^ly  the  remainder  hy  the  sum 
levied  on  each  article. 

2.  What  is  the  duty,  at  $1.25  a  yard,  on  65  pieces  of 
brocade  silk,  each  containing  50  yards  ? 

3.  What  is  the  duty  on  87  hhds.  of  molasses,  at  20  cts. 
per  gallon,  the  leakage  being  3^  ? 

4.  What  is  the  duty,  at  6  cts.  a  pound,  on  500  bags  of 
coffee,  each  weighing  6d>  lbs.,  the  tare  being  2%  ? 

PROBLEM    II. 

462.  To  calculate  Ad    Valorcni  Dufia^f  the   Cost  of  the 

goods  and  the  Rate  being  given. 

5.  What  is  the  ad  valorem  duty,  at  2^%,  on  a  quantity 
of  silks  invoiced  at  $3500  ? 

Analysis. — Since  the  duty  is  25%  ad  valorem,  $35°° 

it  is  .25  times  the  invoice;  and  $3500 x  .25  =  1875.  .25 

Hence,  the  J^^s.  $875.00 

KuLE. — Multiply  the  cost  of  the  goods  hy  the  given  rate, 
expressed  decimally. 

6.  What  is  the  duty,  at  2,zi%  ^^  valorem,  on  1575  yds. 
of  carpeting  invoiced  at  $1.80  per  yard? 

7.  What  is  the  ad  valorem  duty,  at  40^,  on  no  chests 
of  tea,  each  containing  67  lbs.,  and  invoiced  at  90  cts.  a 
pound,  the  tare  being  9  lbs.  a  chest  ? 

461.  How  calculate  specific  duties  when  the  quantity  of  goods  and  the  sum 
evieJ  on  eac  h  article  are  given  ?    462.  How  ad  valorem  ? 


INTEE]S"AL    EEYE]^UE. 

463.  Intei^nal  Hevenue  is  the  income  of  the 
Government  from  Excise  Duties,  Stamp  Duties,  LicenseSj 
Special  Taxes,  Income  Taxes,  etc. 

464.  Excise  Duties  are  taxes  upon  certain  home  productions, 
and  are  computed  at  a  given  per  cent  on  their  value. 

465.  Stamp  Duties  are  taxes  upon  written  instruments ;  as, 
notes,  drafts,  contracts,  legal  documents,  patent  medicines,  etc. 

466.  ^  License  Tax  is  the  sum  paid  for  permission  to  pursue 
certain  avocations. 

467.  Special  Taxes  are  fixed  sums  assessed  upon  certain 
articies  of  luxury ;  as,  carriages,  billiard  tables,  gold  watches,  etc. 

Income  Taxes  are  those  levied  upon  annual  incomes. 

Note. — In  determining  income  taxes,  certain  deductions  are  made 
for  house  rent.  National  and  State  taxes,  losses,  etc. 

468.   To  compute  Income  Taxes,  the  Rate  being  given. 

1.  What  is  a  man's  tax  whose  income  is  $5675 ;  the 
rate  being  5^,  the  deductions  $1000  for  house  rent,  $350 
national  tax,  and  $1 100  for  losses  ? 

Analysis. — Total  deductions  are  $1000 +$350 +  $1100= $2450; 
and  $5675  — $2450= $3225  taxable  income.  Now  5%  of  $3225  = 
3225  X  .05  =  $161. 2 5,  the  tax  required.     Hence,  the 

EuLE. — From  the  ioicome  subtract  the  total  deductions, 
and  multiply  the  remainder  hy  the  rate. 

Note. — If  there  are  special  taxes  on  articles  of  luxury,  as  carriages, 
etc.,  they  must  be  added  to  the  tax  on  the  income. 

2.  What  was  A's  revenue  tax  for  1869,  at  5^ ;  his  income 
being  $4750;  bis  losses  $1185,  and  exemptions  $1200  ? 

3.  In  1870,  A  had  $10500  income;  35  oz.  taxable  plate 
at  5  cts.,  I  watch  $2,  and  i  carriage  I2 :  what  was  his  tax 
at  K%,  allowing  $2100  exemption  ? 


463.  What  is  internal  revenue?  464.  Excise  duties?  465.  Stamp  duties? 
466.  A  license  tax?  467.  Special  taxes?  Income  taxes?  468.  How  compute 
income  taxes,  when  the  rate  is  given  ? 


EQUATION"    OF    PAYMENTS. 

469.  Equation  of  l*ayments  is  finding  the  at-em^^ 
time  for  payment  of  two  or  more  sums  due  at  different  times. 

The  average  time  sought  is  often  called  tiie  mean,  or 
equated  time. 

470.  Equation  of  Payments  embraces  two  classes  of  examples, 
ist.  Those  in  which  the  items  or  bills  have  the  same  date,  but 

different  lengths  of  credit.     2d.  Those  in  which  they  have  different 
dates,  and  the  same  or  different  lengths  of  credit. 

PROBLEM    I. 

471.  To  find  the  Avei'age  Tinier  when  the  Items  have  the 

same  date,  but  different  lengths  of  credit. 

I.  Bought  Oct.  3d,  1870,  goods  amounting  to  the  fol- 
lowing sums:  $50  payable  in  4  m.,  $70  in  6  m.,  and  $80  in 
8  m. :  what  is  the  average  time  at  which  the  whole  may  be 
J)aid,  without  loss  to  either  party? 

Analysis. — The  int.  on  $50  for  4  m.=the  int.  ^i-q  >^  4=200 
on  $1  for  50  times  4  m,  or  200  m.  Again  the  int.  mq  ^  6  =  420 
on  $70  for  6  m.=the  int.  on  $1  for  70  times  6  m.,        80  X  8  =  640 

or  420  m.     Finally,  the  int.  on  $80  for  8  m.=tlie      — — 

int.  of  $1  for  80  times  8  m.,  or  640  m.     Now  200  ^°°   )l2bo 

m. +420  m.  +  640  m.  =  i26o  m. ;  therefore  I  am  A.ns.  6-j^  m. 
entitled  to  the  use  of  $1  for  1260  m.  But  the  sum 
of  the  debts  is  $50 +  $70 +  $80=  $200.  Now  as  I  am  entitled  to  the 
use  of  $1  for  1260  m.,  I  must  be  entitled  to  the  use  of  $200  for  7^0  part 
of  1260  m.,  and  1260-5-200=6 1\  m.,  or  6  m.  9  d.,  the  average  time  re- 
quired.    Hence,  the 

Rule. — Multiply  each  item  ly  its  length  of  credit,  and 
divide  the  sum  of  the  products  ly  the  sum  of  the  items. 
The  quotient  will  he  the  average  time. 

Notes. — i.  When  the  date  of  the  payment  is  required,  add  the 
average  time  to  the  date  of  the  transaction.  Thus,  in  the  preceding 
Ex.,  the  date  of  payt.  is  Oct.  3d  +  6  m.  9  d.,  or  April  12th,  1871. 

469.  What  is  equation  of  payments  ?  Note.  What  is  the  average  time  called  ? 
470.  How  many  classes  of  examples  in  Equation  of  Payments?  471.  How  find 
the  avera^re  time,  when  the  items  have  the  same  date,  but  different  credita 


302  EQUATION     OF     PAYMENTS. 

2.  This  rule  is  applicable  to  notes  as  well  as  accounts.  It  is 
founded  upon  tlie  supposition  that  hank  discount  is  the  same  as 
simple  interest.     Though  not  strictly  accurate,  it  is  in  general  use. 

3.  If  one  item  is  cash,  it  has  no  ^m6,-and  no  p7'oduct;  but  in  find- 
ing the  sum  of  items,  this  must  be  added  with  the  others. 

4.  In  the  answer,  a  fraction  less  than  |  day,  is  rejected ;  if  |  day 
or  more,  i  day  is  added. 

2.  Bought  a  house  June  20th,  1870,  for  $3000,  and  agreed 
to  pay  I  down;  |  in  6  m.,  the  balance  in  12  m.:  at  what 
date  may  the  whole  be  equitably  paid  ? 

3.  A  owes  B  $700,  payable  in  4  mos. ;  $500,  in  6  mos. : 
$800,  in  10  mos.;  and  $1000  in  12  mos.:  in  what  time 
may  the  whole  be  justly  paid? 

4.  Bought  a  bill  of  goods  March  loth,  1868,  amounting 
to  $2500;  and  agreed  to  pay  $500  cash,  $750  in  10  days, 
$600  in  20  days,  $400  in  30  days,  and  $250  in  40  days. 
At  what,  date  may  I  equitably  pay  the  whole  ? 

5.  A  jobber  sold  me  on  the  ist  of  March  I12000  worth 
of  goods,  to  be  paid  for  as  follows :  J  cash,  |  in  2  mos.,  J  in 
4  mos.,  and  the  remaining  J  in  6  mos.  When  may  I  pay 
the  whole  in  equity  ? 

PROBLEM    II. 

472.    To   find   the   Average    Time,   when   the   items   have 
diiferent  dates,  and  the  same  or  different  lengths  of  credit. 

6.  Bought  the  following  bills  of  goods:  March  loth, 
1870,  $500  on  2  m.;  April  4th,  $800  on  4  m.;  June  15th, 
$1000  on  3  m.     What  is  the  average  time? 

ANALYSI8.-We  first  find  May  10,  00  X  $500=  0000  d. 
when  the  items  are  due  by     ^  ^^     g^  ^    ^800=    68800  d. 

adding  the  time  of  credit  to     g^   ^^         128  X  liooo^  128000  d. 

the  date  of  each,  and  for  con-  -^^ , — — —  , 

venience  place  these  dates  ^2300   )  196800(1. 

in  a  column.    Taking  the  Average  time,  85^1  d. 

earliest  date  on  which  either 

item  matures  as  a  standard,  we  find  the  number  of  days  from  this 
standard  date  to  the  maturity  of  the  other  items,  and  place  them  on 
the  ri^ht,  with  the  sign  (  x  )  and  the  items  opposite. 


EQUATIOK     OF     PAYMENTS.  303 

Multiplying  each  item  and  its  number  of  days  together,  the  sum 
of  the  products  shows  that  the  interest  on  the  several  items  is  equal 
to  the  interest  of  $i  for  196800  days. 

Now  if  it  takes  196800  days  for  $1  to  gain  a  certain  sum,  it  will 
take  $2300,  j-h-u  of  196800  d.  to  gain  this  sum ;  and  196800-^-2300= 
85H,  or  86  days  from  the  assumed  date.  Now  May  10  +  86  d.=Aug. 
4th,  the  date  when  the  amt.  is  equitably  due.    Hence,  the 

Rule. — I.  Find  the  date  when  the  several  items  hecome 
due,  and  set  them  in  a  column. 

Take  the  earliest  of  these  dates  as  a  standard,  and  set  the 
number  of  days  from  this  date  to  the  maturity  of  the  other 
items  in  another  column  on  the  right,  with  the  items  opposite. 

II.  Multiply  each  item  ly  its  number  of  days,  and  divide 
the  sum  of  the  lyroducts  by  the  sum  of  the  items.  The 
quotient  will  be  the  average  time  of  credit. 

NoTES.-^i.  Add  the  average  time  to  the  standard  date,  and  the 
result  will  be  the  equitable  date  of  payment. 

2.  The  latest  date  on  which  either  item  falls  due  may  also  be 
taken  as  the  standard ;  and  having  found  the  average  time,  subtract 
it  from  this  date  ;  the  result  will  be  the  date  of  payment. 

3.  The  date  at  which  each  item  becomes  due,  is  readily  found  by 
adding  its  time  of  credit  to  the  date  of  the  transaction. 

7.  A  bought  goods  as  follows:  Apr.  lotli,  $310  on  6  m. ; 
May  2ist,  I468  on  2  m. ;  June  ist,  $520  on  4  m. ;  July  8th, 
$750  on  3  m.:  what  is  the  average  time,  and  at  what  date 
may  the  whole  debt  be  equitably  discharged  ? 

Suggestion. — The  second  item  matures  earliest;  this  date  is 
therefore  the  standard.     Ans.  Av.  time  59  d. ;  date  Sept.  i8tli. 

8.  Sold  the  following  bills  on  6  months  credit:  Jan. 
15th,  $210;  Feb.  nth,  $167;  March  7th,  $320.25  ;  April 
2d,  $500.10:  when  may  the  whole  be  paid  at  one  time? 

9.  Bought  June  5th,  on  4  m.,  groceries  for  I125 ;  June 
/ist,  $230.45;  July  12th,  I267;  Aug.  2d,  $860.80:  what 
is  the  amt.  and  time  of  a  note  to  cover  the  whole  ? 


472.  How  find  the  average  time,  when  the  items  have  different  dates,  and  the 
Bane  or  differeiit  creaits?  Note.  How  find  the  date  of  the  payment?  How  find 
the  date  at  which  each  item  becomes  due  ? 


104 


AVERAGING     ACCOUNTS. 


AVERAGING     ACCOUNTS. 

473.  An  Account  is  a  record  of  the  items  of  debii 
and  credit  in  business  transactions. 

Note. — The  term  debit  is  from  tlie  Latin  debitus,  owed. 

474.  A  3Ierchandise  JBalauce  is  the  difference 
between  the  debits  and  credits  of  an  account. 

A  Cash  balance  is  the  difference  between  the  debits 
and  credits,  with  the  interest  due  on  each. 

Notes. — i.  Bills  of  goods  sold  on  time  are  entitled  to  interest 
after  they  become  due  ;  and  payments  made  before  they  are  due  are 
also  entitled  to  interest. 

475.  To  find  the  Average  Time  for  paying  the  balance  of 
an  Account  which  has  both  debits  and  credits. 

I.  Find  the  cash  Bal.  of  the  following  Acct.,  and  when  due. 
Dr.     Wm.  Gokdon  in  Acct.  with  John  Raistdolph.      Cr. 


1870. 
Feb.  10. 
May  II. 
July  26. 

ForMdse.,  4  m. 

"        "      2  " 

$450.00 
500.00 
360.00 

1870. 
Mar.  20. 
iJuly    9. 
jSept.  15. 

By  Sundries,  3  ni. 
"    Draft,      60  d. 
"   Cash, 

$325-00 
150.00 
400.00 

Analysis.  —  Setting 
down  the  date  when  each 
item  is  due, take  June  loth, 
the  earliest  of  these  dates, 
as  the  standard,  find  the 
number  of  days  from  this 
standard  to  the  maturity 
of  each  item  on  both  sides, 
and  place  it  on  the  right 
of  its  date  with  the  sign 
(  X ),  then  the  items,  add- 
ing 3  days  grace  to  the 
time  of  the  draft. 


Debits. 

June  10,    00  d.  X  I450  =  00000  d. 

Aug.  II,  62  d.  X  500  =:  31000  d. 

Sept.  26,  108  d.  X  360  =■  38880  d. 

Amt.  debits,  $1 3 10,  Int.  69880  d. 

Credits. 

June  20,  10  d.  X  $325  —    3250  d. 

Sept.  10,  92  d.  X  150  =  13800  d. 

"      15,    97  d.  X   400  =  38800  d. 

Amt.  credits,  $875,  Int.  5 5 850  d. 

Cash  bal.  $435,   "   14030  d. 

$435)14030  d.  =  32  d.+. 

Ans.  Bal.  I435,  due  July  12,  '70. 


473.  What  is  an  account?     Note.  What  is  the  meaning  of  the  term  debit! 
474   What  is  a  merchandise  balance  ?    A  cash  balance  ? 


AVERAGING     ACCOUNTS.  305 

Muldplying  each  item  on  both  sides  by  its  number  of  days, 
the  int.  on  the  debits  is  equal  to  the  int.  of  $i  for  69880  days, 
and  the  int.  on  the  credits  is  equal  to  the  int.  of  $1  for  55850  days. 
(Art.  474.)  The  balance  of  int.  on  the  Dr.  side  is  69880  d.  — 55850  d. 
=  14030  d.;  that  is  to  the  int.  of  $1  for  14030  d.  The  balance  of 
items  on  the  Dr.  side  is  $1310— $875  =  1435. 

Now  if  it  takes  14030  days  for  $1  to  gain  a  certain  sum,  it  will 
take  I435,  4^5  of  14030  d.  and  14030^435=321?,  or  32  d.  But  the 
assumed  standard  June  ioth  +  32  d.=July  12th.  Therefore  the 
balance  due  Randolph  the  creditor  is  $435,  payable  July  12th,  1870. 

If  the  greater  sum  of  items  and  the  greater  sura  of  products  were 
«n  opposite  sides,  it  would  be  necessary  to  subtract  the  average  time 
from  the  assumed  date.     Hence,  the 

Rule. — I.  Set  down  the  date  luhen  each  item  of  deUt  and 
credit  is  due;  and  assuming  the  earliest  of  these  dates  as  a 
standard,  ivrite  the  number  of  days  from  this  standard 
date  to  the  maturity  of  the  respective  ite7ns,  on  the  right, 
with  the  sign  (  x )  atid  the  items  themselves  ojjposite. 

II.  Multiply  each  item  hy  its  numher  of  days,  and  divide 
the  difference  between  the  sums  of  products  by  that  between 
the  sums  of  items;  the  quotient  tvill  be  the  average  time. 

III.  If  the  greater  sum  of  items  and  the  greater  sum  of 
products  are  both  on  the  same  side,  add  the  average  time  to 
the  assumed  date;  if  on  opposite  sides,  subtract  it;  and 
the  result  tvill  be  the  date  when  the  balance  of  the  account  is 
equitably  due. 

Notes. — i.  The  average  time  may  be  such  as  to  extend  to  a  date 
either  earlier  or  later  than  that  of  any  of  the  items.  (Ex.  2.) 

2.  In  finding  the  maturity  of  notes  and  drafts,  3  days  grace  yhoulJ 
be  added  to  the  specified  time  of  payment. 

3.  In  finding  the  extension  to  which  the  balance  of  a  debt  ia 
entitled,  when  partial  payments  are  made  before  it  is  due, 

Multiply  each  payment  by  the  time  from  its  date  to  the  maturity 
of  the  debt,  and  divide  the  sum  of  the  products  X)y  the  halance 
remaining  unpaid. 

4.  When  no  time  of  credit  is  mentioned,  the  transaction  is  under- 
stood to  be  for  cash,  and  its  payment  due  at  once. 

475.  How  find  the  average  time  for  paying  the  balance,  when  there  are  both 
debits  and  credits? 


BOG 


AVERAGIl^Q     ACCOUKTl 


2.  Find  the  balance  of  the  following  Acct.,  and  when  due. 
Dr.  A.  B.  in  account  with  C.  D.  Cr. 


i860. 

i860. 

Aug.  II. 

For  Mdse., 

$160.00 

Sept.  2. 

By  Sundries, 

$7500 

Sept.    5. 

«        « 

240.00 

Oct.  10. 

"    your  Note  on  30  d.. 

100.00 

Oct.   20. 

"    I  Horse, 

175.00 

Nov.  I. 

"    Cash 

110.00 

Dr, 

OPKBATION. 

Or, 

Due. 

Bays, 

Items. 

Products. 

Due. 

Days. 

Items. 

Products. 

Aug.  II. 

0 

S160 

0000 

Sept.    2. 

22  - 

$75 

1650 

Sept.    5. 

25 

240 

6000 

Nov.    9. 

93 

100 

9300 

Oct.    20. 

70 

175 

12250 

Nov.    I. 

82       no 

9020 

575 

18250 

285 

19970 

_2^1 

1720-^290=6  d.  (nearly),  av.  time. 

18250 

Bal=$290 

Aug.  1 1 —6  d. = Aug.  5,  bal.  is  due. 

1720 

Suggestion. — In  this  example,  the  greater  sum  of  items  and  the 
greater  sum  of  products  are  on  opposite  sides ;  hence,  the  average 
time  must  be  subtracted  from  the  standard  date. 

3.  A  man  bought  a  cottage  for  $2750, payable  in  i  year; 
in  3  mos.  he  paid  $500,  and  3  mos.  later  I750 :  to  what 
extension  is  he  entitled  on  the  balance  ? 

Solution. — The  sum  of  the  product  is  9000 ;  and  the  balance  of 
the  debt  $1 500.  Now  9000-^  1 500=6.  Ans.  6  mos.  after  the  maturity 
of  the  debt.  (Note  3.) 

4.  A  merchant  sold  a  bill  of  $4220  worth  of  goods  on 
8  mos.;  2  months  after  the  customer  paid  him  $720,  one 
month  later  $850,  and  2  months  later  liooo:  how  long 
should  the  balance  in  equity  remain  unpaid  ? 

5.  What  is  the  balance  of  the  following  account,  and 
when  is  it  due  ? 

Br.    HEifliBT  Swift  in  Acct.  with  Homee  Morgan.    Cr. 


1865. 

1865. 

March  10. 

For  Sundries, 

$250 

April     I. 

By  Bal.  of  Aoct., 

$iiol 

April  15. 

"    Flour  on  60  d., 

420 

May     21. 

"   Dft.onaod., 

3001 

June    20. 

"    Mdse.  on  30  d,, 

boo 

iJuly      I. 

«   Cash, 

560 

1 

RATIO. 

476.  Matio  is  the  relatmi  which  one  number  has  to 
another  with  respect  to  magnitude. 

The  Terms  of  a  ratio  are  the  numbers  compared. 
They  are  often  called  a  couplet. 

477.  Ratio  is  commonly  expressed  by  a  colon  ( : )  placed  between 
tlie  two  numbers  compared.    Thus,  the  ratio  of  6  to  3  is  written  6:3. 

478.  The  jirst  term  is  called  the  antecedent,  the  second  t!.o 
consequent. 

The  comparison  is  made  by  considering  what  multijple  ot  part  the 
antecedent  is  of  the  consequent. 

Notes. — i.  The  dgn  of  ratio  (:)  is  derived  from  tlio  sign  cf 
division  (-f-),  the  horizontal  line  being  dropped. 

2.  The  terms  are  so  called  from  the  order  of  their  position.  They 
must  be  of  the  same  kind  or  denomination;  otherwise  they  cannot 
be  compared. 

479.  Ratio  is  measured  by  9>.  fraction,  the  numerator  of  which  is 
the  antecedent,  and  the  denominator  the  consequent ;  or  what  is  the 
same  thing,  by  dividing  the  antecedent  by  the  consequent. 

480.  The  xialue  of  a  ratio  is  the  value  of  the  fraction  by  which 
it  is  measured.  Thus,  comparing  6  with  2,  we  say  the  ratio  of  6 :  2 
is  f ,  or  3.  That  is,  the  former  has  a  magnitude  which  contains  the 
latter  3  times ;  therefore  the  value  of  the  ratio  6:2  is  3. 

481.  A  Simple  Matio  is  one  which  has  but  two 
terms ;  as,  8 : 4. 

A  Compound  Matio  is  the  product  of  tivo  or  more 
simple  ratios. 

Thus :  4 :  2  1^  are  simple  But  4x9:2x3 

9:33      ratios.  is  a  compound  ratio. 

Note. — ^The  nature  of  compound  ratios  is  the  same  as  that  of 
simple  ratios.  They  are  so  called  to  denote  their  origin,  and  are 
usually  expressed  by  writing  the  corresponding  terms  of  the  simple 
ratios  one  under  another,  as  above. 

476.  What  is  ratio?  The  numbera  compared  called?  477.  How  is  ratio  com- 
monly expressed  ?  The  first  terra  called ?  The  second?  Note.  Why ^  479.  How 
ig  ratio  mcaijured  ?    Tho  value  of  a  ratio  ?    481.  Simple  ratio?    Cosapoundt 


SOS  RATIO. 

482.  Ratio  is  also  distinguished  as  direct  and  inverse  or  reciprocal 

A  direct  ratio  is  one  which,  arises  from  dividing  the  antecedent  by 
the  consequent. 

An  inverse  ratio  is  one  which  arises  from  dividing  the  consequent 
\>j  the  antecedent,  and  is  the  same  as  the  ratio  of  the  reciprocals 
of  the  two  numbers  compared.  (Art.  io6.)  Thus,  the  direct  ratio 
of  4  to  i2=:-j4j,  or  i" ;  the  inverse  ratio  of  4  to  I2=\%  or  3.  It  is  the 
same  as  the  ratio  of  their  reciprocals,  \  to  -^2. 

483.  1'1^6  ratio  between  two  fractions  having  a  common  denomi- 
nator is  the  same  as  the  ratio  of  their  numerators.  Thus,  the  ratio 
of  I :  I  is  the  same  as  6 ;  3. 

In  finding  the  ratio  of  two  fractions  which  have  different  denomi- 
nators, they  should  be  reduced  to  a  common  denominator ;  then  take 
the  ratio  of  their  numerators.     (Art.  153.) 

484.  Ill  finding  the  ratio  between  two  compound  numbers,  they 
must  be  reduced  to  the  same  dcTiomination. 

Note. — Finding  the  ratio  between  two  numbers  is  the  same  in 
principle  as  finding  what  part  one  is  of  the  other,  the  number 
denoting  the  part  being  the  antecedent.  Thus,  2  is  |  of  4— | ;  and 
the  ratio  of  2  to  4  is  |=i.     (Art.  173.) 

485.  Since  ratios  are  measured  by  fractions  whose 
numerators  are  the  antecedents,  and  denominators  the 
consequents,  it  follows  that  operations  have  the  same  effect 
upon  the  terms  of  a  ratio  as  upon  the  terms  of  a  fraction. 
(Art.  144.)     That  is, 

1.  Multiplyi7ig  tlie  antecedent  multiplies  the  ratio;  and 
dividing  the  antecedent  divides  the  ratio. 

2.  Multiplying  the  consequent  divides  the  ratio;  and 
dividing  the  consequent  midtiplies  the  ratio. 

3.  Midtiplying  or  dividing  hoth  the  antecedent  and  con- 
sequent hy  the  same  numher,  does  not  alter  the  value  of  the 
ratio. 

482.  Direct?  Inverse?  483.  The  ratio  of  two  fractions  having  a  common 
denominator  ?  484.  In  finding  the  ratio  of  two  compound  numbers,  Avhat  must 
be  done  ?  485.  What  effect  do  operations  on  the  terms  of  a  ratio  have  ?  Multi- 
plying the  antecedent  ?  Dividing  it  ?  Multiplying  the  consequent  ?  Dividing  it  ? 
■\7hnt  efTcct  has  multiplying  or  dividing  both  the  antecedent  and  consequent  by 
the  tame  number? 


PROPORTION.  309 

What  are  the  ratios  of  the  following  couplets: 

1.  12  14.  4.  6  ;  24.  7.  £5  :  los.  6d. 

2.  28:7.  5.  8  :  40.  8.  10  y.  :  6  ft.  3  in. 
3.36:12.             6.9:51.  9.  25  g.  :  2  qt.  I  pt. 

10.  Eednce  the  ratio  of  14  to  35  to  the  lowest  terms  ? 
Solution. — 14 :  35  equals  ^ ;  and  i*=f,  or  2  :  5.    (Art.  146.) 
Reduce  the  following  ratios  to  the  lowest  terms  ? 

11.  154:28.         13.  73:511.  15.  238:1428. 

12.  39:165.         14.  113:1017.         16.  576:1728. 

17.  Reduce  the  ratio  f  :|-  to  the  lowest  integral  terms? 
f =il.  and  f =if.     Now  ^f  :  1^-  is  the  same  as  4  :  5.    (Art.  483.) 

18.  Multiply  the  ratio  of  2 1 :  7  by  4 :  8.  Ans.  S4:  ^6,  or  i^ 

19.  What  is  the  value  of  5:8x4:  10x7:9? 


PROPORTION. 

486.  JPropoi^tion  is  an  equality  of  ratios. 

487.  Every  proportion  must  have  at  least  four  terms', 
for,  the  equality  is  between  tivo  or  more  ratios,  and  each 
ratio  has  two  terms,  an  antecedent  and  a  consequent. 

A  proportion  may,  however,  be  formed  from  three  num- 
hers;  for,  one  of  the  numbers  may  be  repeated,  so  as  to 
form  two  terms. 

488.  Proportion  is  denoted  in  two  ways ;  by  a  double 
colon  ( : : ),  and  by  the  sign  of  equality  (  =  ),  placed  between 
the  ratios.  Thus,  each  of  the  expressions  4 :  2  : :  6 : 3,  and 
4:2  =  6:3  indicates  a  proportion ;  for,  f =f. 

The  former  is  read,  "  4  is  to  2  as  6  to  3,"  or  "  4  is  the 
same  part  of  2,  that  6  is  of  3."  The  latter  is  read,  "  the 
ratio  of  4  to  2  equals  the  ratio  of  6  to  3." 

Note. — The  sign  ( : : )  is  derived  from  the  sign  (=),  the  points 
being  the  extremities  of  the  parallel  lines. 

486.  What  is  proportion  ?  487.  How  many  terms  has  every  proportion  ?  Caa 
three  mimbers  form  a  proportion  ?    How  ? 


310  PKOPOTITIOK. 

489.  The  four  numbers  which  form  a  proportion,  are 
called  proportionals.  The  first  and  last  are  the  extremes^ 
the  other  two  the  ineans. 

When  a  proportion  has  but  three  numbers,  the  second 
term  is  called  a  mean  proportional  between  tlie  other  two. 

490.  If  four  numbers  are  proportional,  the  product  of 
the  extremes  is  equal  to  the  product  of  the  means.     Hence, 

491.  The  relation  of  the  four  terms  of  a  proportion  to 
each  other  is  such,  that  if  any  three  of  them  are  giyen,  the 
other  or  missing  term  may  be  found. 

492.  To  find  the  3Iissing  Term  of  a  Proportion,  the  other 
three  Terms  being  given, 

1.  Let  6  be  the  first  term  of  a  proportion,  3  and  10  the  two 
means;  tlie  other  extreme  equals  3xio-r-6=5;  for,  the  product 
of  the  meansr=the  product  of  the  extremes;  and  the  product  of  two 
factors  divided  by  one  of  them,  gives  the  other.     (Art.  93.) 

2.  Let  3,  10  and  5  be  the  last  three  terms  of  a  proportion,  the  first 
term  equals  3  x  10-^-5  =  30-7-5  or  6. 

3.  Let  6  and  5  be  the  extremes  of  a  proportion,  and  3  one  of  the 
means  ;  the  other  mean  equals  6  x  5-7-3=30-^3  or  10. 

4.  If  6  and  5  are  the  extremes,  and  10  one  of  the  means,  the  other 
mean  equals  6  x  5-7-10=30^10  or  3.     Hence,  the 

EuLE, — I.  If  one  of  the  extremes  and  the  two  means  are 
given,  divide  the  product  of  the  means  hy  the  given  extreme, 

II.  If  one  of  the  means  and  the  tivo  extremes  are  given, 
divide  the  product  of  the  extremes  hy  the  given  mean. 

Find  the  missing  term  in  the  following  proportions : 

1.  52  :  13  1:62:  — .  5.  4  rods  :  11  ft. : :  18  men :  — . 

2.  15:90::  — :  72.  6.  24  yd. :3  yd. ::  —  :$i2. 

3.  60  :  — : :  100  :  33J.         7.  20  gal. :  — : :  $40  :  $8. 

4.  _:  25  ::  |:  I;.  8.  — :  40  lb.: :  £2  :  8s. 

4Sg.  What  are  the  four  numbers  forming  a  proportion  called?  491.  If  three 
terms  of  a  proportion  are  given,  what  is  true  of  the  fourth  ?  If  the  two  means 
and  one  extreme  are  given,  how  find  the  other  extreme  ?  If  the  two  extremes 
and  one  mean  are  given,  how  find  the  other  mean  ? 


SIMPLE    PROPOETIO]^. 

493.  Simple  I^ropor^tion  is  an  equality  of  two 
simple  ratios. 

Simple  JProportion  is  applied  cliiefly  to  the  solu- 
tion of  problems  having  three  terms  given  to  find  a,  fourth, 
of  which  the  third  shall  be  the  same  multiple  or  part,  as 
the  Jlrst  is  of  the  second. 

494.  To  solve  Problems  by  Simple  Proportion. 

1.  If  5  baskets  of  peaches  cost  $io,  what  will  3  baskets  cost? 
Analysis. — The  question  assumes  statement. 

that  3  baskets  can  be  bought  at  the  5  bas. :  3  bas. ::  ^10  :  Ans. 
same  rate  as  5  baskets ;  therefore  3 

5  bas.  has  the  same  ratio  to  3  bas.  S)'!^© 

as  the  cost  of  5  bas.  has  to  the  cost  of  ^y    j 

3  bas.    That  is,  5  bas. :  3  bas. : :  $10 : 

cost  of  3  bas.  We  have  then  the  two  means  and  one  extreme  of  a 
proportion  to  find  the  fourth  term,  or  other  extreme.  (Art.  492.) 
Now  the  product  of  the  means  $10  x  3  =  $3o;  and  $30-7-5  (the  other 
extreme) =$6,  the  cost  of  3  baskets.     Hence,  the 

Rule. — I.  Tahe  that  numler  for  the  third  term^  which  is 
the  same  hind  as  the  answer. 

II.  When  the  answer  is  to  he  larger  than  the  third  term, 
place  the  larger  of  the  other  two  numbers  for  the  second 
term;  but  when  less,  place  the  smaller  for  the  second  tertn, 
and  the  other  for  the  first. 

III.  Multiply  the  second  and  third  terms  together,  and 
divide  the  product  by  the  first ;  the  quotient  will  be  the 
fourth  term  or  answer.     (Arts.  490,  491.) 

Proof. — If  the  product  of  the  first  and  fourth  terms 
equals  that  of  the  second  and  tliird,  the  answer  is  right. 

Notes. — i  The  arrangement  of  the  given  terms  in  the  form  of  a 
proportion  is  called  "  Stating  the  question." 

2.  After  stating  the  question,  the  factors  common  to  the  first  and 
§€cond,  or  to  the  first  and  third  terms,  should  be  cancelled. 

493.  Simple  Proportion  ?    494.  How  aolve  problems  by  Simple  Proportion  ? 


C12  SIMPLE     PKOPOIlTIOiT. 

3.  If  tlie  first  and  second  terms  contain  different  denominations^ 
tliey  must  be  reduced  to  the  same.  If  the  tliird  term  is  a  compound 
number,  it  must  be  reduced  to  the  lowest  denomination  it  contains. 

495.  Reasons, — i.  The  reason  for  placing  that  number  for  the 
tliird  term,  which  is  the  same  kind  as  the  answer,  and  the  other  two 
numbers  for  the  first  and  second,  is  because  money  has  a  ratio  to 
money,  but  not  to  the  other  two  numbers ;  and  the  other  two  numbers 
have  a  ratio  to  each  other,  but  not  to  money. 

2.  Of  the  two  like  numbers,  the  smaller  is  taken  for  the  second 
term,  and  the  larger  for  the  first,  because  3  baskets  being  less  than 
5  baskets,  will  cost  less ;  consequently,  the  answer  or  fourth  term 
must  be  less  than  the  third,  the  cost  of  5  baskets. 

3.  If  it  were  required  to  find  the  cost  of  a  quantity  greater  than 
that  whose  cost  is  given,  the  answer  would  be  greater  than  the  third 
term ;  consequently  the  greater  of  the  two  similar  numbers  must 
then  be  taken  for  the  second  term,  and  the  less  for  the  first. 

4.  The  reason  for  multiplying  the  second  and  third  terms  together 
and  dividing  the  product  by  the  first,  is  because  the  product  of  the 
means  divided  by  one  of  the  extremes,  gives  the  other  extreme  ot 
ansioer.    (Arts.  93,  492.) 

2.  If  9  yards  of  cloth  cost  $54,  what  will  23  yards  cost? 
Statement.— g  yd. :  23  yd. : :  $54 :  Ans. 

And  ($54  X  23)-^9=$i38,  the  Ans. 

By  Cancellation. — Since  9  yds.  cost  Operation. 

$54,  I  yd.  will  cost  ^  of  $54,  or  $Y ;  ^V"  ^  2^  =  A7is. 

and  23  yds.  will  cost  $  V"  x  23=ii-^i^       6,  5^  x  2  3      ^      „     , 

9  '- =^  =  $138,  ^ns. 

=$13S,  Ans.  ^  '^ 

Proof.— <)  yd. :  23  yd. : :  $54  :  $138 ;  for  9  x  138=23  x  54. 

3.  If  7  barrels  of  flour  cost  $56,  what  will  20  barrels  cost? 

4.  What  cost  75  bushels  of  wheat,  if  15  bushels  cost  %;^;^? 

5.  What  cost  150  sheep,  if  17  sheep  cost  $51  ? 

6.  If  5  lb.  8  oz.  of  honey  cost  $1.65,  what  will  20  lb.  cost? 

7.  Paid  £1,  15s.  6d.  for  6  pounds  of  tea:  what  must  be 
paid  for  a  chest  containing  65  lb.  8  oz.  ? 

495.  Why  take  the  number  which  is  of  the  same  kind  as  the  answer  for 
the  third  term,  and  the  other  two  for  the  first  and  second  ?  When  place  tlie 
larger  of  the  other  two  numbers  for  the  second?  Why?  When  the  smaller? 
Why  ?  Why  does  the  product  of  the  second  and  third  terms  divided  by  the  first 
give  the  answer?  What  is  the  arrangement  of  the  temis  in  the  form  of  a  propor- 
tion called  ?  If  the  first  and  second  terms  contain  different  denominations,  hoT^ 
proceed  ?    If  the  third  is  a  compound  nnmber,  how  ? 


SIMPLE     PROPORTIOJ^.  313 


SIMPLE    PROPORTION    BY    ANALYSIS. 

496.  The  chief  diflBculty  experienced  bj  the  pupil  in  Simple 
Proportion,  lies  in  "  stating  the  question."  This  difficulty  arises 
from  a  want  of  familiarity  with  the  relation  of  numbers.  He  will  be 
assisted  by  analyzing  the  examples  before  attempting  to  state  them. 

8.  If  7  hats  cost  $42,  liow  mucli  will  12  hats  cost? 
Analysis. — i  hat  is  i  seventh  of  7  hats ;   therefore  i  hat  will 

cost  I  seventh  as  much  as  7  hats  ;  and  \  of  $42  is  $6.  Again,  12  hats 
will  cost  12  times  as  much  as  i  hat,  and  12  times  $6  are  $72.  There- 
fore, 12  hats  will  cost  $72. 

Or,  7  hats  are  f^  of  12  hats ;  therefore  the  cost  of  7  hats  is  -^^  the 
cost  of  12  hats.  But  7  hats  cost  $42 ;  hence  $42  are  -,^2  the  cost  of 
12  hats.  Now,  if  -f^-  of  a  number  are  $42,  tV  of  that  number  is  |  of 
$42,  which  is  $6 ;  and  ||  are  12  times  $6,  or  $72. 

By  Proportion.— 'J  h. :  12  h. : :  $42  :  Ans.     That  is, 
7  h.  are  the  same  part  of  12  h.  as  $42  are  of  the  cost  of  12  hats. 

497.  Solve  the  following  examples  both  by  analysis  and 
proportion, 

9.  If  II  men  can  cradle  33  acres  of  grain  in  i  day,  how 
many  acres  can  45  men  cradle  in  the  same  time  ? 

10.  When  mackerel  are  $150  for  12  barrels,  what  must 
I  pay  for  75  barrels  ?  ^       ^ 

11.  How  far  will  a  railroad  car  go  in  12  hours,  if  it 
goes  at  the  rate  of  15  miles  in  40  minutes  ? 

12.  A  bankrupt  owes  $3500,  his  assets  are  $1800:  how 
much  will  a  creditor  receive  whose  claim  is  $560  ? 

13.  At  the  rate  of  18  barrels  for  %62„  what  will  235  bar- 
rels of  apples  cost  ? 

14.  If  $250  earns  I17J  interest  in  i  year,  how  much 
will  1 1 900  earn  in  the  same  time? 

15.  If  a  car  wheel  turns  round  6  times  in  T^^t  yards,  how 
many  times  will  it  turn  round  in  going  7  miles  ? 

16.  If  a  clerk  can  lay  up  1 1500  in  i^  year,  how  long  will 
It  take  him  to  lay  up  $5000  ? 

17.  If  6  men  can  hoe  a  field  of  corn  m  20  hours,  how 
long  will  it  take  1 5  men  to  hoe  it  ?  c 

14 


314  SIMPLE     PROPOETIOK. 

1 8.  An  engineer  found  it  would  take  75  men  220  days 
to  build  a  fort;  the  general  commanding  required  it  to  be 
built  in  15  days:  how  many  men  must  the  engineer 
smploy  to  complete  it  in  the  required  time  ? 

19.  If  5  oz.  of  silk  can  be  spun  into  a  thread  100  rods 
long,  what  weight  of  silk  is  required  to  spin  a  thread  that 
will  reach  the  moon,  240000  miles  distant  ? 

20.  How  many  horses  will  it  take  to  consume  a  scaffold 
of  hay  in  40  days,  if  12  horses  can  consume  it  in  90  days  ? 

21.  If  I  acre  of  land  costs  $15,  what  will  2^-}  acres  cost  ? 

22.  If  f  of  a  ton  of  iron  costs  £f,  what  will  -^^  of  a  ton  cost  ? 
2^.  If  loj  lb.  sugar  cost  $1  J,  what  will  3of  lb.  cost  ? 

24.  If  I  of  a  chest  of  tea  costs  $35.50,  what  will  15^ 
chests  cost  ? 

25.  What  will  48!  tons  of  hay  cost,  if  1 2^  tons  cost  |i  26  J  ? 
2  6.  If  y'2  of  a  ship  is  worth  $  1 6  2  5  of,  what  is  t\  of  it  worth  ? 

27.  What  will  it  cost  me  for  a  saddle  horse  to  go  100 
miles  if  I  pay  at  the  rate  of  $7^  cts.  for  3  miles  ? 

28.  What  must  be  the  length  of  a  slate  that  is  10  in. 
wide,  to  contain  a  square  foot  ? 

29.  How  many  yards  of  carpeting  J  yard  wide  will  it 
take  to  cover  a  floor  15  ft.  long  and  12  feet  wide  ? 

30.  If  a  man's  pulse  beats  68  times  a  minute,  how  many 
times  will  it  beat  in  24  hours  ? 

31.  If  Halley's  comet  moves  2°  45'  in  11  hours,  bow  far 
will  it  move  in  30  days  ? 

"32.  If  a  pole  10  ft.  high  cast  a  shadow  7^  ft.  long,  how 
high  is  a  flag-staff  whose  shadow  is  60  ft.  long? 

SS.  At  the  rate  of  3  oranges  for  7  apples,  how  many 
oranges  can  be  bought  with  150  apples? 
'-    34.  If  an  ocean  steamer  runs  1250  miles  in  3  days  8  h., 
how  far  will  she  run  in  8  days  ? 

35.  If  12  men  can  harvest  a  field  of  wheat  in  11  days, 
how  many  men  are  required  to  harvest  it  in  4  days  ? 

36.  The  length  of  a  croquet-ground  is  45  feet ;  and  lU 
width  is  to  its  length  as  2  to  3  :  what  is  its  width  ? 


Simple    propoktioi^.  ?15 

37.  A  man's  annual  income  from  U.  S.  6s  is  $1350  when 
gold  is  112J  :  whafc  was  it  when  gold  was  160  ? 

^8.  George  has  10  minutes  start  in  a  foot-race;  and 
runs  20  rods  a  minute:  how  long  will  it  take  Henry,  who 
runs  28  rods  a  minute,  to  overtake  him? 

39.  If  the  driving  wheel  of  a  locomotive  makes  227 
revolutions  in  going  206  rods  6  ft.,  how  many  revolutions 
will  it  make  in  running  18  miles  240  rods? 

40.  If  3  lbs.  of  coffee  cost  $1.20,  and  10  lbs.  of  coffee  are 
worth  6  lbs.  of  tea,  what  w^ill  60  lbs.  of  tea  cost  ? 

41.  A  can  chop  a  cord  of  wood  in  4  hours,  and  B  in  6 
hours :  how  long  will  it  take  both  to  chop  a  cord  ? 

42.  A  reservoir  has  3  hydrants;  the  first  will  empty  it 
in  8  hours,  the  second  in  10;  the  third  in  12  hours:  if  all 
run  together,  how  long  will  it  take  to  empty  it  ? 

43.  A  man  and  wife  drank  a  keg  of  ale  in  18  d. ;  it  would 
last  the  man  30  d. :  how  long  would  it  last  the  woman  ? 

=  44.  A  fox  is  100  rods  before  a  hound,  but  the  hound 
runs  20  rods  while  the  fox  runs  18  rods:  how  far  must 
the  hound  run  before  he  catches  the  fox  ? 

45.  A  cistern  holding  3600  gallons  has  a  supply  and  a 
discharge  pipe ;  the  former  runs  45  gallons  an  hour,  the 
latter  ^3  gallons :  how  long  will  it  take  to  fill  the  cistern, 
when  both  are  running  ? 

46.  A  clerk  who  engaged  to  work  for  I500  a  year,  com- 
menced at  12  o'clock  Jan.  ist,  1869,  and  left  at  noon,  the 
2ist  of  May  following:  how  much  ought  he  to  receive  ? 

47.  A  church  clock  is  set  at  12  o'clk.  Saturday  night; 
Tuesday  noon  it  had  gained  3  min. :  what  will  be  the  true 
time,  when  it  strikes  9  the  following  Sunday  morning  ? 

48.  A  market-woman  bought  109  eggs  at  2  for  a  cent, 
and  another  100  at  3  for  a  cent;  if  she  sells  them  at  the 
rate  of  5  for  2  cents,  what  will  she  make  or  lose  ? 

49.  Two  persons  being  :^:^6  miles  apart,  start  at  the  same 
time,  and  meet  in  6  days,  one  traveling  6  miles  a  day 
faster  than  the  other:  how  far  did  each  travel ? 


OOMPOUl^D    PEOPORTIOK 


*     >  : :  1 2  :  2,  is  a  componnd  proportion. 


498.    Compound  JProportioii  is  an  equality  of 
a  som2)Ound  and  a  simjjle  ratio.     Thus, 
4: 
9 

It  is  read,  '*  The  ratio  of  4  into  9  is  to  2  into  3  as  12  to  2." 

I.  If  4  men  saw  20  cords  of  wood  in  5  days,  how  many 
Cards  can  1 2  men  saw  in  3  days  ? 

Analysis. — The  answer  is   to  statement. 

be  in  cords;    we  therefore  make      4  m.  :12  m.  )    .  o  >  A^q 

20  c.  the  third  term.     The  other     5  d.  :  3  d.      [   *  * 
p:iven  numbers  occur  in  pairs,  two      20  C.  X  12  X3  =  720C. 
of  a  kind ;  as,  4  men  and  12  men,  4x5  =  20 

5  days  and  3  days.     We  arrange  720  C.-j-20  =  36  C.  Ans. 

these  pairs  in  ratios,  as  we  should 

in  simple  proportion,  if  the  answer  depended  on  each  pair  alone. 
That  is,  since  12  m.  will  saw  more  than  4  men,  we  take  12  for  the 
second  term  and  4  for  the  first.  Again,  since  12  men  will  saw  less 
in  3  days  than  in  5  days,  we  take  3  for  the  second  term  and  5  for  the 
first.  Finally,  dividing  the  product  of  the  second  and  third  terms 
20  c.  X  12  X  3  =  720  c.  by  the  product  of  the  first  terms  4  x  5  =  20,  wo 
have  36  cords  for  the  answer.     Hence,  the 

EuLE. — I.  Mahe  that  number  the  third  term  which  is  of 
the  same  Icind  as  the  ansiuer. 

II.  TalcG  the  other  numbers  in  pairs  of  the  same  Tcind, 
and  arrange  them  as  if  the  answer  depended  on  each  couplet, 
as  in  simple  proportion.     (Art.  494.) 

III.  Multiply  the  second  and  third  terms  together,  and 
divide  the  product  hy  the  product  of  the  first  terms,  cancelling 
the  factors  common  to  the  first  and  second,  or  to  the  first 
and  third  terms.     Tlie  quotient  will  he  the  ansiver. 

Proof. — If  the  product  of  the  first  and  fourth  terms 
equals  that  of  the  second  and  third  terms,  the  luorh  is  right, 

498.  What  is  Compound  Proportion  ?  Explain  the  first  example.  The  rule. 
Proof.  Note.  How  proceed  when  tlie  firpt  and  second  terms  contain  different 
denominations  ?  When  the  third  does  ?  How  else  may  questions  in  Compound 
Proportion  be  solved  ? 


COMPOUND     PROPORTIO]^.  317 

Notes. — i.  The  terms  of  eacli  couplet  in  fhe  compound  ratio  must 
be  reduced  to  the  same  denomination,  and  the  third  term  to  the  lowest 
denomination  contained  in  it,  as  in  Simple  Proportion, 

2.  In  Compound  Proportion,  all  the  terms  are  given  in  couplets  or 
pairs  of  the  same  kind,  except  one.  This  is  called  the  odd  term,  or 
demand,  and  is  always  the  same  kind  as  the  answer. 

3.  It  should  be  observed  that  it  is  not  the  ratio  of  4  to  2,  nor  of  9  to 
3  alone  that  equals  the  ratio  of  12  to  2 ;  for,  4-5-2  =  2  and  9-5-3=3, 
while  i2-j-2=6.  But  it  is  the  ratio  compounded  of  4  x  g  to  2x3, 
which  equals  the  ratio  of  12  to  2.  Thus,  (4X  g)-=-(2  x  3)=i6 ;  and 
12-^2=6.     (Art.  498.) 

4.  Compound  Proportion  was  formerly  called  "  Double  Rule  of 
Three." 

499.  Problems  in  Compound  Proportion  may  also  be  solved  by 
Analysis  and  Simple  Proportion.     Take  the  preceding  example  : 

By  Analysis. — If  4  men  saw  20  cords  in  5  d.,  i  m.  will  saw  \  of 
20  c,  which  is  5  c,  and  12  m.  will  saw  12  times  5  c,  or  60  cords,  in 
the  same  time.  Again,  if  12  men  saw  60  c.  in  5  days,  in  i  d.  they 
will  saw  \  of  60  c ,  or  12  cords,  and  in  3  d.  3  times  12  c,  or  36  cords, 
the  answer  required. 

By  Simple  Proportion. — 4  m. :  12  m. : :  20  c. :  the  cords  12  m.  will 
saw  in  5  d. ;  and  20  c.  x  12-5-4=60  c.  in  5  days.  Again,  5  d. :  3  d. : : 
60  c. :  the  cords  12  men  will  saw  in  3  d.  And  60  c.  x  3^5  =  36  cords, 
the  same  as  before. 

2.  If  4  men  earn  $219  in  30  clays,  working  10  hours  a 
day,  how  muoli  can  9  men  earn  in  40  days,  working 
8  hours  a  day  ? 

sTATEMEiirT.  By  CanceUatkm. 


S^m.  3 

^Bd. 
B.h.  4 
: :  $219  :  Anfi, 


4  m.:  9  m.   )  4  m. 

30  d.  :  40  d.  V  : :  $2 19  :  A7is.         %,  15,  SH  d. 
loh.  :8h.    )  ■         \%h. 

{I219  X  9  X40  X  8)-^  

(4  X  30  X  10)  =  $5251,  ^Iws.  5       15219x4x3=1: 

$5251,  J^^5. 

3.  If  6  men  can  mow  28  acres  in  2  days,  how  long  will 
it  take  7  men  to  mow  42  acres  ? 

4.  If  8  horses  can  plow  32  acres  in  6  days,  how  many 
hcrses  will  it  take  to  plow  24  acres  in  4  days  ? 

5.  If  the  board  of  a  family  of  8  persons  amounts  to  $300 
in  15  weeks,  how  long  will  $1000  board  12  persons? 


318  COMPOUND     PROPORTIO]!^. 

6.  What  will  be  the  cost  of  28  boxes  of  candles  contain - 
'ng  20  pounds  apiece,  if  7  boxes  containing  15  pounds 
ij)iece  can  be  bought  for  $23.75  ? 

7.  If  the  interest  of  $300  for  10  months  is  I20,  what 
will  be  the  interest  of  $1000  for  15  months? 

8.  If  a  man  walks  180  miles  in  6  days,  at  10  h.  each, 
how  many  miles  can  he  walk  in  15  days,  at  8  h.  each  ? 

9.  If  it  costs  $160  to  pave  a  sidewalk  4  ft.  wide  and 
40  ft.  long,  what  will  it  cost  to  pave  one  6  ft.  wide  -md 
125  ft.  long? 

10.  If  it  requires  800  yards  of  cloth  f  yd.  wide  to  supply 
100  men,  how  many  yards  that  is  f  wide  will  it  require  to 
clothe  1500  men? 

11.  If  75  men  can  build  a  wall  50  ft.  long,  8  ft.  high,  and 
3  ft.  thick,  in  10  days,  how  long  will  it  take  100  men  to 
build  a  wall  150  ft.  long,  10  ft.  high,  and  4  ft.  thick  ? 

12.  If  it  costs  $56  to  transport  7  tons  of  goods  no  miles, 
how  much  will  it  cost  to  transport  40  tons  500  miles  ? 

13.  If  30  lb.  of  cotton  will  make  3  pieces  of  muslin  42  yds. 
long  and  f  yd.  wide,  how  many  pounds  will  it  take  to 
make  50  pieces,  each  containing  35  yards  1}  yd.  wide  ? 

14.  If  the  interest  of  $600,  at  7^,  is  $35*  for  10  months, 
what  Avill  be  the  int.  of  $2500,  at  6'/c,  for  5  months  ?.  '/' 

i$.  If  9  men,  working  10  hours  per  day,  can  make  18 
sofas  in  30  days,  how  many  sofas  can  50  men  make  in 
90  days,  working  8  hours  per  day  ?  i>''  ^ 

16.  If  it  takes  9000  bricks  8  in.  long  and  4  in.  wide  to 
pave  a  court-yard  50  ft.  long  by  40  ft.  wide,  how  many 
tiles  10  in.  square  will  be  required  to  lay  a  hall-floor  75  ft 
long  by  8  ft.  wide  ?  ." 

17.  If  in  8  days  15  sugar  maples,  each  running  12  quarts 
of  sap  per  day,  make  10  boxes  of  sugar,  each  weighing 
6  lb.,  how  many  boxes  weighing  10  lb.  apiece,  will  a  maple 
grove  containing  300  trees,  make  in  36  days,  each  tree 
running  16  quarts  per  day?  A71S.  720  boxes. 


PARTITIVE     PROPORTION.  319 

PARTITIVE    PROPORTION. 

500.  Partitive  Proportion  is  dividing  a  mimber 
into  two  or  more  parts  having  a  given  ratio  to  each  other. 

501.    To    divide   a    Number   into    two    op    more    parts  which 
shall  be  proportional  to  given  numbers. 

1.  A  and  B  found  a  purse  of  money  containing  $35, 
which  they  agree  to  divide  between  them  in  the  ratio  of  2 
to  3 ;  how  many  dollars  will  each  have  ? 

By  Proportion.— The  sum  of  the  statement. 

proportional  parts  is  to  each  separate  5  :  2  :  :  $35  :  A's  S. 

part  as  the  number  to  be  divided  is  to  -  .  ^  .  .  |„-  .  j^jg  g^ 

each  man's  share.     That  is,  5  (2  +  3) 

is  to  2  as  $35  to  A's  share.     Again,  5      W5  X2)h-5=$I4Ass. 
is  to  3  as  $35  to  B's  share.    Hence,  the     ($35X3)^5  =  ^21  B's  S. 

Exile. — I.  Take  the  miynber  to  he  divided  for  the  third 
term;  each  proj^ortional  part  successively  for  the  seco7id 
term;  and  their  sum  for  the  first. 

XL  The  product  of  the  second  and  third  terms  of  each 
proportion,  divided  by  the  first,  will  he  the  corresponding 
part  required. 

By  Analysis. — Since  A  had  2  parts  and  B  3,  both  had  2  +  3,  or  5 
parts.     Hence,  A  will  have  |  and  B  \  of  the  money.     Now  %  of  $35 

=:$I4,  and  \  of  $35 =$21.     Hence,  the 

EuLE. — Divide  the  given  number  by  the  sum  of  the  pro- 
portional "^umbers,  and  multiply  the  quotient  by  each  one's 
jjroportional  part. 

2.  It  is  required  to  divide  78  into  three  parts  Vfhich 
shall  be  to  each  other  as  3,  4,  and  6.     Ans.  18,  24,  2,^. 

3.  A  man  having  200  sheep,  wished  to  divide  them 
into  three  flocks  which  should  be  to  each  other  as  2,  3, 
and  5  :  how  many  Avill  each  flock  contain  ? 

4.  A  miller  had  250  bushels  of  provender  composed  of 
oats,  peas,  and  corn  in  the  proportion  of  3,  4,  and  5 1 :  how 
many  bushels  were  there  of  each  kind  ? 

500.  How  divide  a  number  into  parts  having  a  given  ratio  ? 


320  PARTNERSHIP. 

5.  A  father  divided  497  acres  of  land  among  his  four 
sons  in  proportion  to  their  ages,  which  were  as  2,  3,  4,  and 
5 :  how  many  acres  did  each  receive  ? 

502.  The  principles  of  Partitive  Proportion  are  appli- 
cable to  those  classes  of  problems  commonly  arranged 
under  the  heads  of  Partnership,  BanTcruptcy,  General 
Average,  etc.,  in  which  a  given  number  is  to  be  divided 
into  parts  having  a,  given  ratio  to  each  other. ' 


PAETNEESHIP. 

503.  JPartnersJiip  is  the  association  of  two  or  more 
persons  in  business  for  their  common  profit. 

It  is  of  two  kinds ;  Simple  and  Compound. 

504.  SiTTiple  I^artnership  is  that  in  which  the 
capital  of  each  partner  is  employed  for  the  same  time. 

505.  Compound  JPartnersJiip  is  that  in  which 

the  capital  is  employed  for  unequal  times. 

Note. — The  association  is  called  a  firniy  Jiouse,  or  company ;  and 
the  persons  associated  are  termed  partners. 

506.  The  Capital  is  the  money  or  property  employed 
in  the  business. 

The  JProfits  are  the  gains  shared  among  the  partners, 
and  are  called  dividends. 

Notes. — i.  The  profits  are  divided  as  the  partners  may  agree. 
Other  things  being  equal,  when  the  capital  is  employed  for  the  same 
time,  it  is  customary  to  divide  the  i)rofits  according  to  the  amount 
of  capital  each  one  furnishes. 

2.  When  the  capital  is  employed  for  unequal  times,  the  profits  are 
usually  divided  according  to  the  amount  of  capital  each  furnisifaes, 
and  the  time  it  is  employed. 

502.  To  what  are  the  principles  of  Partitive  Proportion  applicable  ?  503.  What 
is  Partnership  ?  Of  how  many  kinds  ?  504.  Simple  Partnership  ?  505.  Com- 
pound ?  Note.  What  is  the  association  called  ?  506.  What  is  the  capital  ?  The 
profltP?  What  called? 


PARTi^EESHIP.  321 

PROBLEM    I. 
507.  To  find  each  Partner's  Share  of  the  Profit  or  Loss, 
when  divided  according  to  their  capital. 

1.  A  and  B  entered  into  partnership ;  the  former  furnish- 
ing $648,  the  latter  $1080,  and  agreed  to  divide  the  profit 
according  to  their  capital.  They  made  I432 :  what  was 
each  one's  share  of  the  profit  ? 

Analysis. — The  capital  equals  $648  +  §io8o=$i728. 

$1728  (capital) :    $648  (A's  cap.) : :  $432  (profit) :  A's  share,  or  $162. 

$1728        "        :  $1080  (B's  cap.) : :  $432      "       :  B's  share,  or  $270. 

Or  thus:  The  profit  $432-7-11728  (the  cap.)=.25  ;  that  is,  the 
profit  is  25^  of  the  capital.  Therefore  each  man's  share  of  the 
profit  is  25%  of  his  capital.  Now  $648  x  .25  =  $162,  A's  share ;  and 
$1080  X  .25  =  $270,  B's  share.     Hence,  the 

EuLE. — The  wJiole  capital  is  to  each  partner'' s  capital,  as 
the  whole  profit  or  loss  to  each  partners  share  of  the  profit 
or  loss. 

Or,  Fhid  ichat  per  cent  the  profit  or  loss  is  of  the  whole 
capital,  and  mtdtiply  each  mail's  capital  hy  it.     (Art.  339.) 

2.  A  and  B  form  a  partnership,  A  furnishing  $1200, 
and  B  $1500;  they  lose  I500:  what  is  each  one's  share  of 
the  loss  ? 

3.  A  and  B  buy  a  saw-mill,  A  advancing  $3000,  and 
B  $4500;  they  rent  it  for  I850  a  year:  what  should  each 
receive  ? 

4.  The  net  gains  of  A,  B,  and  C  for  a  year  are  $12500; 
A  furnishes  $15000,  B  $12000,  and  0  $10000 :  how  should 
the  profit  be  divided  ? 

5.  Three  persons  entering  into  a  speculation,  made 
$15300,  which  they  divided  in  the  ratio  of  2,  3,  and  4: 
how  much  did  each  receive  ? 

6.  A,  B,  and  C  hire  a  pasture  for  $320  a  year;  A  put  in 
80,  B  120,  and  0  200  sheep:  how  much  ought  each  man 
to  pay  ? 

507.  How  find  each  partner's  profit  or  loss,  when  their  capital  is  employed  the 
same  time  ? 


OA.2  PARTI^EKSHIP. 


PROBLEM    II 


508.  To  find  each   Partner's  Share  of  the  Profit  or  Loss, 
when  divided  according  to  capital  and  time. 

7.  A  and  B  enter  into  partnership ;  A  furnishes  $400 
for  8  months,  and  B  $600  for  4  months;  they  gain  $350: 
what  is  each  one's  share  of  the  profit  ? 

Analysis. — In  this  case  the  profit  of  the  partners  depends  on  two 
conditions,  viz. :  the  amount  of  capital  each  furnishes,  and  the  time 
it  is  employed. 

But  the  use  of  $400  for  8  months  equals  that  of  8  times  $400,  or 
$3200,  for  I  m. ;  and  $600  for  4  m.  equals  4  times  $600,  or  $2400,  for 
I  m.  The  respective  capitals,  then,  are  equivalent  to  $2400  and 
$3200,  each  employed  for  i  m.  Now,  as  A  furnished  $3200,  and 
B  $2400,  the  whole  capital  equals  $3200+  $2400= $5600.    Therefore, 

$5600  :  $3200  : :  $350  (profit) :  A's  share,  or  $200. 
$5600  :  $2400  : :  $350      "       :  B's  share,  or  $150. 

Or  thus:  The  gain  $350-7-15600  (the  cap.)  =  .06^,  or  6}%. 
Therefore,  $32oox  .o6i=$20o,  A's  share;  and  $240ox  .o6l  =  $i50, 
B's  share.     Hence,  the 

EuLE. — 3fuUiply  each  partner^s  capital  ly  the  time  it  is 
employed.  Consider  these  products  as  their  respective  capi- 
tals, and  proceed  as  in  the  last  problem. 

Or,  Fi7id  ivhat  per  cent  the  profit  or  loss  is  of  the  ivhole 
capital,  and  multiply  each  man's  capital  hy  it.     (Art.  339.) 

Note. — The  object  of  multiplying  each  partner's  capital  by  the 
time  it  is  employed  is,  to  reduce  their  respective  capitals  to  equivalents 
for  the  same  time. 

8.  A,  B,  and  C  form  a  partnership ;  A  furnishing  I500 
for  9  m.,  B  I700  for  i  year,  and  0  $400  for  15  months; 
they  lose  $600 :  what  is  each  man's  share  of  the  loss  ? 

9.  Two  men  hire  a  pasture  for  $50 ;  one  put  in  20  horses 
for  12  weeks,  the  other  25  horses  for  10  weeks:  how  much 
should  each  pay  ? 


508.  How  find  each  one's  share,  when  their  capital  is  employed  for  unequal 
timet?  ?    Note.  Why  multiply  each  one's  capital  by  the  time  it  is  employed? 


BAN-KRUPTCY.  323 

10  A.,  B,  C,  and  J)  commenced  business  Jan.  ist,  1870, 
when  A  Airnisiied  $1000;  March  ist,  B  put  in  $1200; 
July  ist,  0  put  in  $1500;  and  Sept.  ist,  D  put  in  $2000; 
during  the  year  they  made  $1450 :  how  much  should  each 
receive  ? 

11.  A  store  insured  at  the  Howard  Insurance  Co.  for 
$3000;  in  the  Continental,  $4500;  in  the  American, 
$6000,  was  damaged  by  fire  to  the  amount  of  $6750: 
what  share  of  the  loss  should  each  company  pay  ? 

Remark. — The  loss  should  be  averaged  in  proportion  to  the  risk 
assumed  by  each  company. 

12.  A  quartermaster  paid  $2500  for  the  transportation 
of  provisions ;  A  carried  150  barrels  40  miles,  B  170  barrels 
60  miles,  C  210  barrels  75  miles,  and  D  250  barrels  100 
miles :  how  much  did  he  pay  each  ? 

13.  A,  B,  and  C,  formed  a  partnership,  and  cleared 
I12000;  A  put  in  I8000  for  4  m.,  and  then  added  $2000 
for  6  m.;  B  put  in  $16000  for  3  m.,  and  then  withdrawing 
half  his  capital,  continued  the  remainder  5  m.  longer; 
C  put  in  $13500  for  7  m. :  how  divide  the  profit. 


ba:nkeuptcy. 

509.  Bankruptcy  is  inability  to  pay  indebtedness. 

Note. — A  person  unable  to  pay  his  debts  is  said  to  be  insolvent, 
and  is  called  a  bankrupt. 

510.  The  Assets  of  a  bankrupt  are  the  property  in 
his  possession. 

The  Liabilities  are  his  debts. 

511.  The  ^et  JProceeds  are  the  assets  less  the  ex- 
pense of  settlement.  They  are  divided  among  the  creditors 
according  to  their  claims. 

509.  "What  is  bankruptcy  ?  ^10.  What  are  assets  ?  Liabilities?  511.  Net  pro- 
ceeds? 512.  How  find  each  creditor's  dividend,  when  the  liabilities  and  the  net 
proceeds  are  given  ? 


324  ALLIGATION". 

512.    To   find  each   Creditor's   Dividend,   the   Liabilities   and 
Net  Proceeds  being  given. 

I.  A  merchant  failed,  owing  B  $1260,  0  $1800,  and 
D  $1940;  his  assets  were  $1735,  and  the  expenses  of 
settling  $435  :  how  much  did  each  creditor  receive  ? 

Analysis. — The  liabilities  are  $1260  +  $1800  +  $1940=15000 ;  and 
the  net  proceeds  $173.5— $435 =$1300.     Now, 

$5000 :  $1260  : :  $1300 :  B's  dividend,  or  $327.60. 

$5000  :  $1800  : ;  $1300  :  C's         "         or  $468.00. 

$5000  :  $1940  :  :^i300  :  D's         "         or  $504.40. 

Or  thus:  The  net  proceeds  $1300-7- $5000=. 26,  or  26%,  the  rate 

he  is  able  to  pay.     (Art.  339.)    Now   $1260  x  .26  =  $327.60  B's ; 

$1800  X  .26= $468  C's ;  $1940  X  .26= $504.40  D's.     Hence,  the 

EuLE. — The  wliole  liabilities  are  to  each  creditor's  claim, 
as  the  net  j^roceeds  to  each  creditors  dividend. 

Or,  Find  what  per  cent  the  net  proceeds  are  of  the 
liabilities,  and  multiply  each  creditors  claim  by  it. 

2.  A  bankrupt  owes  A  $6300,  B  I4500,  and  D  $3200; 
his  assets  are  $5250,  and  the  expenses  of  settling  $1500: 
how  much  will  each  creditor  receive  ? 

3.  A.  B.  &  Co.  went  into  bankruptcy,  owing  $48400, 
and  having  $13200  assets;  the  expense  of  settling  was 
$1 100.     What  did  D  receive  on  $8240  ? 


ALLIGATIOlsr. 

513.  Alligation  i^oiU^o'kmdi^, Medial d^ndi Alternate. 

Alligation  Medial  is  the  method  of  finding  the 
mean  value  of  mixtures. 

Alligation  Alternate  is  the  method  of  finding 
the  proportional  parts  of  mixtures  having  a  given  value. 

Notes. — i.  Alligation,  frran  the  Latin  alligo,  to  tie,  or  hind 
together,  is  so  called  from  the  manner  of  connecting  the  ingredients 
by  curve  lines  in  some  of  the  operations. 

2.  Alternate,  Latin  alternatus,  by  turns,  refers  to  the  manner  of 
connecting  the  prices  above  the  mean  price  with  those  helow. 

3.  The  term  medial  is  from  the  Latin  mMius,  middle  or  average. 

513.  What  ie  alligation  medial  ?    Alternnt-e? 


ALLIGATION.  32^ 


ALLIGATION     MEDIAL. 

514.  To  find  the  3Iean  Value  of  a  mixture,  the  Price  and 
Quantity  of  each  ingredient  being  given. 

1.  Mixed  50  lbs.  of  tea  at  90  cts.,  60  lbs.  at  $1.10,  and 
80  lbs.  at  $1.25  :  what  is  the  mean  yalue  of  the  mixture  ? 

Analysis.— The  total  value  of  the  first  $.90  X  50=  $45.00 
kind  .90x50 =$45. GO,  the  second  $1.10x60  $1.10x60;=  $66,00 
=z$66.oo,  the  third  $1.25  x  8o=$ioo.oo  ;  $1.25  X  8o  =  $ioo.oo 
therefore  the  total  value  of  the  mixture  =  f^  \  $2  1 1. 00 

$211 .00,    But  tlie  quantity  mixed=  190 lbs.  a        ^^  ^^ 

Now,  if  190  lbs.  are  worth  $211,  i  lb.  is 

worth  r^o  of  $211,  or  $i.n  +  .     Therefore  the  mean  value  of  the 
mixture  is  $1.11  +  per  pound.     Hence,  the 

EuLE. — Divide  the  value  of  the  whole  mixture  ly  the  sitm 
of  the  ingredients  mixed. 

Note. — If  an  ingredient  costs  nothing,  as  water,  sand,  etc.,  its 
value  is  o ;  but  the  quantity  itself  must  bo  added  to  the  other 
ingredients. 

2.  If  I  mix  3  kinds  of  sugar  worth  12,  15,  and  20  cts.  a 
pound,  what  is  a  pound  of  the  mixture  worth  ? 

3.  A  farmer  mixed  30  bu.  of  corn,  at  $1.25,  with  25  bu. 
of  oats,  at  60  cts.,  and  10  bu.  of  peas,  at  95  cts. :  what  was 
the  average  yalue  of  the  mixture  ? 

4.  A  grocer  mixed  5  gal.  molasses,  worth  80  cts.  a  gal., 
and  107  gallons  of  water,  with  a  hogshead  of  cider,  at  20 
cts. :  what  was  the  average  worth  of  the  mixture  ? 

5.  A  goldsmith  mixed  12  oz.  of  gold  22  carats  fine  with 
8  oz.  20  carats,  and  7  oz.  18  carats  fine:  what  was  the 
average  fineness  of  the  composition  ? 

6.  A  milkman  bought  40  gallons  of  new  milk,  at  4  cts, 
a  quart,  and  60  gallons  of  skimmed  milk  at  2  cts.  a  quart, 
which  he  mixed  with  1 2  gallons  of  water,  and  sold  tho 
whole  at  6  cts.  a  quart :  required  his  profit  ? 

Note.  Why  is  this  rule  called  alligation  ?  Why  alternate  ?  W^hat  is  the  import 
of  medial?  514.  How  find  the  mean  value  of  a  mixture,  when  the  price  and 
quantity  are  given  ? 


3-^  3  lb. 

11^  J  I  " 


326  ALLIGATION     ALTERNATE. 

ALLIGATION     ALTERNATE. 

PROBLEM    I. 

515.    To  find  the  I*ro2)ortloiial  Parts  of  a    Mixture,  the 
Mean  Price  and  the  Price  of  each  ingredient  being  given. 

7.  A  grocer  desired  to  mix  4  kinds  of  tea  worth  3s.,  8s., 
IIS.,  and  i2S.  a  pound,  so  that  the  mixture  should  be  worth 
9s.  a  pound:  in  what  proportion  must  they  be  taken? 

Analysis. — To  equalize  tlie  gain  and  loss,  we  opekation. 

compare  the  prices  in  pairs,  one  being  aboce  and 
the  other  helow  the  mean  price,  and,  for  conve- 
nience, connect  them  by  curve  lines.  Taking  the 
first  and  fourth;  on  i  lb.  at  3s.  the  gain  is  6s. ; 
on  I  lb.  at  i2S.  the  loss  is  3s.,  which  we  place  op- 
posite the  12  and  3.  Therefore,  it  takes  i  lb.  at  3s.  to  balance  the 
loss  on  2  lb.  at  I2S.,  and  the  proportional  parts  of  this  couplet  are 
as  I  to  2,  or  as  3  to  6.  But  3  and  6  are  the  differences  between  the 
mean  price  and  that  of  the  teas  compared,  taken  inversely. 

Again,  i  lb.  at  8s.  gains  is.,  and  i  lb.  at  us.  loses  2s.,  which  we 
place  opposite  the  11  and  8.  Therefore,  it  takes  2  lb.  at  8s.  to  bal- 
ance the  loss  on  i  lb.  at  us.,  and  the  proportional  parts  of  this 
couplet  are  as  2  to  i.  But  2  and  i  are  the  differences  between  the 
mean  price  and  that  of  the  teas  compared,  taken  inversely.  The 
parts  are  3  lbs.  at  3s.,  2  lbs.  at  8s.,  i  lb.  at  us.,  6  lbs.  at  12s. 

If  we  compare  i\\e  first  and  third,  the  second  and  fourth,  the  pro- 
portional parts  will  be  2  lbs.  at  3s.,  3  lbs.  at  8s.,  6  lbs.  at  us.,  and 
I  lb.  at  I2S.     Hence,  the 

EuLE. — I.  Write  the  prices  of  the  ingredients  in  a  columUy 
with  the  mean  price  on  the  left,  and  talcing  thenn  in  pairs, 
one  less  and  the  other  greater  than  the  mean  price,  connect 
them  hy  a  curve  line. 

II.  Place  the  difference  hetiveen  the  mean  price  and  that 
of  each  ingredient  opposite  the  price  with  which  it  is  com- 
p>ared.  The  sum  of  the  differences  standing  opposite  each 
price  is  the  proportional  part  of  that  ingredient. 

515.  How  And  the  proportional  parts  of  a  mixture,  wheii  the  mean  price  aud 
the  price  of  each  ingredient  are  given  ? 


ALLIGATION"     ALTERNATE.  327 

Rem. — Since  tlie  results  sliow  tlie  proportional  parts  to  be  taken, 
it  follows  if  eacli  is  multijjlied  or  divided  by  the  same  number,  an 
endless  variety  of  answers  may  be  obtained. 

2D  Method. — Since  the  mean  price  is  gs.  a  pound, 

/   I  lb.  at  3S.  gains  6s. ;  hence,  to  gain  is.  takes  ^  lb.=^.  lb. 
)   I  lb.  at  8s.  gains  is. ;  hence,  to  gain  is.  takes  i  lb.=f  lb. 
J   I  lb.  at  IIS.  loses  2s. ;  hence,  to  lose  is.  takes  ^  lb.=|  lb. 
(   I  lb.  at  I2S.  loses  3s. ;  hence,  to  lose  is.  takes  ^  lb.  =  |  lb. 
Reducing  these  results  to  a  common  denominator,  and  using  the 
numerators,  the  proportional  parts  are  i  lb.  at  3s.,  6  lbs.  at  8s., 
3  lbs.  at  IIS.,  and  2  lbs.  at  12s.    Hence,  the 

Rule. — Take  the  given  prices  in  pairs,  one  greater,  the 
other  less  than  the  mean  price,  and  find  how  much  of  each 
article  is  required  to  gain  or  lose  a  unit  of  the  mean  price, 
setting  the  result  on  the  right  of  the  corresponding  price. 

Reduce  the  results  to  a  co7nmo7i  denominator,  and  the 
numerators  will  be  the  proportional  parts  required. 

Notes. — i.  If  there  are  three  ingredients,  compare  the  price  of 
the  one  which  is  greater  or  less  than  the  mean  price  with  each  of 
the  others,  and  take  the  sum  of  the  two  numbers  opposite  this  price. 

2.  The  reason  for  considering  the  ingredients  in  pairs,  one  alove, 
and  the  other  heloic  the  mean  price,  is  that  the  loss  on  one  may  be 
counterbalanced  by  the  gain  on  another. 

[For  Canfield's  Method,  see  Key  to  New  Practical.] 

8.  A  miller  bought  wheat  at  I1.60,  $2.10,  and  I2.25  per 
bushel  respectively,  and  made  a  mixture  worth  $2  a  bushel: 
how  much  of  each  did  he  buy  ? 

9.  A  refiner  wished  to  mix  4  parcels  of  gold  15,  18,  21, 
and  22  carats  fine,  so  that  the  mixture  might  be  20  carats 
fine :  what  quail tity  of  each  must  he  take  ? 

10.  A  grocer  has  three  kinds  of  spices  worth  32,  40,  and 
45  cts.  a  pound :  in  what  proportion  must  they  be  mixed, 
that  the  mixture  may  be  worth  38  cts.  a  pound? 

11.  A  grocer  mixed  4  kinds  of  butter  worth  20  cts., 
27  cts.,  35  cts.,  and  40  cts.  a  pound  respectively,  and  sold 
the  mixture  at  42  cts.  a  pound,  whereby  he  made  10  cts. 
a  pound :  how  much  of  a  kind  did  he  mix  ? 


12  — 

^    3  lb. 

14- 

^15" 

21  — 

^]6" 

23- 

^  4" 

328  ALLIGATIOi^     ALTERJS^ATE. 


PROBLEM    II. 

516.  When  one  Ingredient,  the  Price  of  each,  and  the  Mean 
Price  of  the  Mixture  are  given,  to  find  the  other  Ingredients. 

12.  How  much  sugar  worth  12,  14,  and  21  cts.  a  pound 
must  be  mixed  with  12  lbs.  at  23  cts.  that  the  mixture 
may  be  worth  18  cts.  a  pound  ? 

Analysis. — If  neither  ingredient  were  limited, 
tlie  proportional  parts  would  be  3  lbs.  at  12  cts., 
5  lbs.  at  14  cts.,  6  lbs.  at  21  cts.,  and  4  lbs.  at 
23  cts. 

But  the  quantity  at  23  cts.  is  limited  to  12  lbs., 
which  is  3  times  its  difference  4  lbs.  Now  the  ratio  of  12  lbs.  to 
4  lbs.  is  3.  Multiplying  each  of  the  proportional  parts  found  by  3 
the  result  will  be  9  lbs.,  15  lbs.,  18  lbs.,  and  12  lbs.     Hence,  the 

EuLE. — Find  the  proportional  parts  as  if  ilie  quantity 
of  neither  ingredient  were  limited.     (Art.  515.) 

Multiply  the  parts  thus  found  ly  the  ratio  of  the  given 
ingredient  to  its  proportional  part,  and  the  products  will 
he  the  corresponding  iyigredients  required. 

Note. — When  the  quantities  of  tico  or  viore  ingredients  are  given, 
find  the  average  value  of  them,  and  considering  their  sum  as  one 
quantity,  proceed  as  above,     (Art.  515.) 

13.  How  much  barley  at  40  cts.,  and  corn  at  80  cts., 
must  be  mixed  with  10  bu.  of  oats  at  30  cts.  and  20  bu. 
of  rye  at  60  cts.,  that  the  mixture  may  be  55  cts.  a  bu.  ? 

Suggestion. — The  mean  value  of  10  bu.  of  oats  at  30  cts.  an(^ 
20  bu.  of  rye  at  60,  is  50  cts.  a  bu.    Ans.  30  bu.  barley,  and  24  bu.  corn. 

14.  How  much  butter  at  40,  45,  and  50  cts.  a  pound 
respectively,  must  I  mix  with  30  lbs.  at  65  cts.  that  the 
mixture  may  be  worth  60  cts.  a  pound  ? 

15.  How  many  quarts  of  milk,  worth  4  and  6  cts.  a  quart 
respectively,  must  be  mixed  with  50  quarts  of  water  s( 
that  the  mixture  may  be  worth  5  cts.  a  quart  ? 

516.  When  one  ingredient,  the  price  of  each  and  mean  price  are  given,  ho\» 
find  the  other  ingredients  ? 


ALLIGATIOIT     ALTER  1^-ATE.  329 


PROBLEM    III. 

517.    To    find    the    Ingredients,    the    Price    of   each,    th® 
Quantity  mixed,  and  the  Mean  Price  being  given. 

1 6.  How  much  water  must  be  mixed  with  two  kinds  of 
Bourbon  costing  $4  and  $6  a  gal.,  to  make  a  mixture  of 
T50  gal.  the  mean  price  of  which  shall  be  I3  a  gal.  ? 

Analysis. — The  price  of  the  water  is  o.  Disregarding  the  quantity 
to  be  mixed,  and  proceeding  as  in  Problem  I.,  the  proportional  part? 
are  4  g.  water,  3  g.  at  $4,  and  3  g.  at  $6,  the  sum  of  which  is  4  g-  ^ 

3  g.  +  3  g.=io  gallons. 

But  the  whole  mixture  is  to  be  150  gallons.     Now  the  ratio  of 
150  g.  to  10  g.  equals  \^^^,  or  15. 
Multiplying  each  of  the  parts  previously  obtained  by  15,  we  have 

4  gal.  X  15=60  gal.  water;  3  gal.  x  15=45  gal.  at  $4;  and  3  gal.  x 
15=45  gal.  at  $6.    Hence,  the 

EuLE. — Find  the  proportional  parts  without  regard  tc 
the  quantity  to  he  mixed,  as  i7i  Problem  I. 

Multiply  each  of  the  proportional  parts  thus  found  hj 
the  ratio  of  the  given  mixture  to  the  sum  of  these  parts,  and 
the  several  products  will  he  the  corresponding  ingredients 
required, 

17.  A  grocer  mixed  100  lb.  of  lard  worth  6,  8,  and  12  cts. 
a  pound,  the  mean  value  of  the  mixture  being  10  cts. : 
how  many  pounds  of  each  kind  did  he  take  ? 

18.  Having  coffees  worth  28,  30,  2>^,  and  42  cts.  a  pound 
respectively,  I  wish  to  mix  200  lbs.  in  such  proportions 
that  the  mean  value  of  the  mixture  shall  be  7,6  cts. 
a  pound :  how  many  pounds  of  each  kind  must  I  take  ? 

19.  A  grocer  wished  to  mix  4  kinds  of  petroleum  worth 
40,  45,  50,  and  60  cts.  a  gal.  respectively:  how  much  of 
each  kind  must  he  take  to  make  a  mixture  of  300  gallons, 
worth  52  cts.  a  gallon? 

SI 7.  How  find  the  ingredientB,  when  the  price  of  each,  the  quantity  mised, 
and  the  mean  price  are  given  ? 


INVOLUTION. 

518.  Involution  is  finding  a  'power  of  a  number. 

A  I^oivev  is  the  product  of  a  number  multiplied  intc 
itself     Thus,  2x2  =  4;  3x3  =  9,  etc.,  4  and  9  are  powers. 

519.  Vo-WQY^  2a'Q  (^lYidLQdiYnio  different  degrees  ;  as,  first, 
second,  tliird,  fourth,  etc.  The  name  shows  lioiv  many 
times  the  number  is  taken  as  a  factor  to  produce  the  power. 

520.  The  First  ^ower  is  the  root  or  number  itself 
The  Second  I^oiver  is  the  product  of  a  number 

taken  twice  as  a  factor,  and  is  called  a  square. 

The  Thi7*d  JPower  is  the  product  of  a  number  taken 
three  tiynes  as  a  factor,  and  is  called  a  cube,  etc. 

Notes. — i.  The  second  power  is  called  a  square,  because  the  pro- 
cess of  raising  a  number  to  the  second  power  is  similar  to  that  of 
finding  the  area  of  a  square.     (Art.  243.) 

2.  The  third  power  in  like  manner  is  called  a  cube,  because  the 
process  of  raising  a  number  to  the  third  power  is  similar  to  that  of 
finding  the  contents  of  a  cube.     (Art.  249.) 

521.  Powers  are  denoted  by  a  small  figure  placed  above  the 
number  on  the  right,  called  the  index  or  exponent ;  because  it  shows 
Iww  many  times  the  number  is  taken  as  a  factor,  to  produce  the  power. 

Note. — The  term  index  (plural  indices),  Latin  indicere,  to  proclaim. 
Exponent  is  from  the  Latin  exponere,  to  represent.    Thus, 

2' ==2,  the  first  power,  which  is  the  number  itself. 

2^=2  X  2,  the  second  power,  or  square. 

2^=2  X  2  X  2,  the  third  power,  or  cube. 

2'*=2  X  2  X  2  X  2,  i\\Q  fourth  power,  etc, 

522.  The  expression  2^  is  read,  "2  raised  to  the  fourth  power,  or 
the  fourth  power  of  2." 

1.  Read  the  following:  9^,  12',  25^,  245^  38110,  465'%  looo^*. 

2.  6^  X  74,  2S^  X  48'^  1408—753,  256^0  -r-  975. 


518.  What  is  involntion  ?  A  power  •'  519.  How  are  powers  divided  ?  520.  What 
is  the  first  power  ?  The  second  ?  Third  ?  Note.  Why  is  the  second  power  called 
a  square?    Why  the  third  a  cube ?    521.  How  are  powers  denoted? 


IJs^VOLUTIOIT.  331 

3.  Express  tlie  4th  power  of  85.  5.  The  7tli  power  of  340. 

4.  Express  the  5th  power  of  348.         6.  The  8th  power  of  561. 

523.  To  raise  a  Number  to  any  required  Power. 

7.  What  is  the  4th  po^v^r  of  3  ? 

Analysts. — The  fourth  power  is  the  product  of  a  number  into 
itself  taken  four  times  as  a  factor,  and  3  x  3  x  3  x  3=81,  the  answer 
required.    Hence,  the 

KuLE. — Multiply  the  number  into  itself,  till  it  is  taken 
as  many  times  as  a  factor  as  there  are  units  in  the  index 
of  the  required  poiver. 

Notes. — i.  In  raising  a  number  to  a  power,  it  should  be  observed 
that  the  number  of  multiplications  is  always  one  less  than  the  number 
of  times  it  is  t^^'^en  as  a  factor;  and  tlierefore  one  less  than  tho 
number  of  the  index.  Thus,  4^=4  x  4  x  4,  the  4  is  taken  three  times 
as  a  factor,  but  there  are  only  two  multiplications. 

2.  A  decimal  fraction  is  raised  to  a  power  by  multiplying  it  into 
itself,  and  pointing  oJ0F  as  many  decimals  in  each  power  as  there  are 
decimals  in  the  factors  employed.     Thus,  .i'  =  .oi,  .22=.oo8,  etc. 

3.  A  common  fraction  is  raised  to  a  power  by  multiplying  each 
term  into  itself.     Thus,  (f)^—  ,Sg. 

4.  A  mixed  number  should  be  reduced  to  an  improper  fraction,  or 
the  fractional  part  to  a  decimal;  then  proceed  as  above.  Thus,  (2|)* 
_(.^)2_iA.  or  2^=2.5  and  (2.5)-2=6.25. 

5.  All  powers  of  i  are  i ;  for  i  x  i  x  i,  etc.  =  i. 

Compare  the  square  of  the  following  integers  and  that 
of  their  corresponding  decimals  ; 

8.  5>  6,  7,  8,  9,  10,  20,  30,  40,  50,  60,  70,  80,  90. 

9.  .5,  .6,  .7,  .8,  .9,  .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09. 

Eaise  the  following  numbers  to  the  powers  indicated : 


10.  53. 

13-  4^. 

16.  2.033. 

19.  r. 

II.    2^. 

14.  8^ 

17.  4.00033. 

20.    13.- 

12.    1323. 

15-  25^ 

18.  400.053. 

21.    2-J4. 

523.  How  raise  a  number  to  a  power?  Note.  In  raising  a  number  to  a  power, 
how  many  multiplications  are  there  ?  How  is  a  decimal  raised  to  a  power  ?  A 
common  fraction?  A  mixed  number?  524.  How  find  the  product  of  two  or 
more  powers  of  the  same  number  ? 


332 


INVOLUTIOiq-. 


524.    To  find  the  Product  of  two  or  more   Powers  of  th© 

same  Number. 

2  2.  What  is  the  product  of  4^  multiplied  by  4^  ? 

Analysis.— 4=^=4 X 4 X 4,  and  42=4x4;  therefore  in  tlie  product 
of  4^  x  4^,  4  is  taken  3  +  2,  or  5  times  as  a  factor.  But  3  and  2  are  the 
given  indices  ;  therefore  4  is  taken  as  many  times  as  a  factor  as  there 
are  units  in  the  indices.  Ans.  4^.    Hence,  the 

EuLE. — Add  the  indices^  and  the  sum  ivill  he  the  index 
of  the  product. 

23.  Mult.  2 3  by  22.  25.  Mult.  4^  by  4^ 

24.  Mult.  3"^  by  3 3.  26.  Mult.  5"^  by  5 2. 


FORMATION     OF     SQUARES. 

525.  To  find  the  Square  of  a   Number  in  the  Terms  of  it* 

Parts. 

I.  Find  the  square  of  5  in  the  terms  of  the  parts  3  and  2. 

Analysis. — Let  the  shaded  part  of  the 
diagram  represent  the  square  of  3  ; — each 
f.ide  being  divided  into  3  inches,  its  con- 
tonts  are  equal  to  3  x  3,  or  9  sq.  in. 

The  question  now  is,  what  additions 
must  be  made  and  how  made,  to  preserve 
the  form  of  this  square,  and  make  it  equal 
to  the  square  of  5. 

1st.  To  preserve  the  form  of  the  square 
it  is  plain  equal  additions  must  be  made 
to  two  adjacent  sides  ;  for,  if  made  on  one  side,  or  on  opposite  sides, 
the  figure  will  no  longer  be  a  square. 

2d.  Since  5  is  2  more  than  3,  it  follows  that  two  rows  of  3  squares 
each  must  be  added  at  the  top,  and  2  rows  on  one  of  the  adjacent 
sides,  to  make  its  length  and  breadth  each  equal  to  5.  Now  2  into  3 
plus  2  into  3  are  12  squares,  or  twice  the  product  of  the  two  parts 
2  and  3. 

But  the  diagram  wants  2  times  2  small  squares,  as  represented  by 
the  dotted  lines,  to  till  the  corner  on  the  right,  and  2  times  2  or  4  is 
the  square  of  the  second  part.  We  have  then  9  (the  sq.  of  the  ist 
})art),  12  (twice  the  prod,  of  the  two  parts  3  and  2),  and  4  (the  square 
of  the  2d  part).     But  94-12  +  4=25,  the  square  required. 


t)yoLUTiOK.  333 

Again,  if  5  is  divided  into  4  and  r,  the  square  of  4  is  16,  twice  the 
prod,  of  4  into  i  is  8,  and  the  square  of  i  is  i.     But  16  +  8  +  1=25. 

2.  Eequired  the  square  of  25  in  the  terms  of  20  and  5. 
Analysis.— Multiplying  20  by  20  gives       2^—    20  +  5 

400  (the  square  of  the  ist  part);   20x5         25        20  +  5 

plus  20  X  5  gives  200  (twice  the  prod,  of  J^     400  +  100 
the  two  parts) ;  and  5  into  5  gives  25  (the  "^      ^       ,  ^^^  ,  ^ 

square  of  the  2d  part).    Now  400  +  200  +  25  - — — - 

=625,or252.    Hence,  universally,  625=400  +  200  +  25 

The  square  of  any  number  expressed  in  the  terms  of  its 
parts,  is  equal  to  tlie  square  of  the  first  part,  plus  twice  the 
product  of  the  two  parts,  plus  the  square  of  the  second  part. 

3.  What  is  the  square  of  23  in  the  parts  20  and  3  ? 

4.  What  is  the  square  of  2^  or  2  +^  ?  Ans.  6|. 

5.  What  is  the  square  of  f  or  f  +  i  ?  Ans,  4, 


EVOLUTION. 

526.  Evolution  is  finding  a  root  of  a  number. 
A  Hoot  is  one  of  the  equal  factors  of  a  number. 

527.  Boots,  h'ke  powers,  are  divided  into  degrees;  as,  the 
square,  or  second  root ;  the  cube,  or  third  root ;  the  fourth 
root,  etc. 

528.  The  Square  Moot  is  one  of  the  two  equal 
factors  of  a  number.  Thus,  5  x  5  =  25  ;  therefore,  5  is  the 
square  root  of  25. 

529.  The  Cube  Moot  is  one  of  the  throe  equal 
factors  of  a  number.  Thus,  3x3x3^27;  therefore,  3  is 
the  cube  root  of  27,  etc. 

525.  To  what  is  the  square  of  a  number  equal  in  the  terms  of  its  part»? 
506.  What  is  evolution  ?    A  root  ?    528.  Square  root  ?    529.  Cube  root  ? 


334  EVOLUTION. 

530.  Roots  are  denoted  in  two  loays:  ist.  By  prefixing 
to  the  number  the  character  (  |/),  called  the  radical  sigyi, 
with  a  figure  over  it ;  as  V4,  \/8. 

2d.  By  a  fractional  exponent  placed  above  the  number 
on  the  right.     Thus,  V9,  or  9*,  denotes  the  sq.  root  of  9. 

Notes. — i.  The  figure  over  the  radical  sign  (|/)  and  the  denomi- 
nator of  the  exponent  respectively,  denote  the  name  of  the  root. 

2.  In  expressing  the  square  root,  it  is  customary  to  use  simply  the 
radical  sign  (-i/),  the  2  being  understood.  Thus,  the  expression 
1/25  =  5,  is  read,  "  the  square  root  of  25  =  5." 

3.  Tlie  term  radical  is  from  the  Latin  radix,  root.  The  sign  (t  ) 
is  a  corruption  of  the  letter  r,  the  initial  of  radix. 

531.  A  Perfect  JPower  is  a  number  whose  exact 
root  can  be  found. 

An  Imperfect  JPower  is  a  number  whose  exact 
root  can  not  be  found. 

532.  A  Surd  is  the  root  of  an  imperfect  power. 
Thus,  5  is  an  imperfect  power,  and  its  square  root  2.23  + 
is  a  surd. 

Note. — All  roots  as  well  n^  powers  of  i,  are  i. 
Eead  the  following  expressions : 

1.  1/40.       3.  119^        5.  i.5i         7.  V256.  9.  Vff. 

2.  V15.       4.  243^-        6.  V29.        8.  V45.7.        10.  V-B|. 
1 1.  Express  the  cube  root  of  64  both  ways ;  the  4th  root 

of  25  ;  the  7th  root  of  81 ;  the  loth  root  of  100. 

533.  To  find   how  many  figures  the  Square  of  a   Number 
contains. 

ist.  Take  i  and  9,  the  least  and  greatest  integer  that  can 
be  expressed  by  one  figure;  also  10  and  99,  the  least  and 
greatest  that  can  be  expressed  by  two  integral  figures,  etc. 
Squaring  these  numbers,  we  hare  for 

The  Roots:       i,  •  9,     10,      99,       100,        999,  etc. 

The  Squares:    i,  81,  100,  9801,  loooo,  998001,  etc. 

530.  How  are  roots  denoted?  531.  A  perfect  power?  Imperfect?  532.  h 
Burd  ?    5»j3.  How  many  figures  has  the  square  of  a  number  ? 


SQUARE     ROOT.  335 

2d.  Take  .  i  and  .9,  the  least  and  greatest  decimals  that 
can  be  expressed  by  one  figure;  also  .01  and  .99,  the  least 
and  greatest  that  can  be  expressed  by  tivo  decimal  figures, 
etc.     Squaring  these,  we  have  for  • 

TheEoots:     .1,    .9,    .01,      .99,      .001,        .999,       etc. 

The  Squares:  .01,  .81,  .0001,  .9801,  .000001,  .998001,  etc. 

By  inspecting  these  roots  and  squares,  we  discover  that 

Tlie  square  of  the  number  contains  twice  as  ma7i7j  figures 
as  its  root,  or  ttuice  as  many  less  one. 

534.    To   find    how   many  figures    the  Square  Root    of  a 
Number  contains. 

Divide  the  number  into  periods  of  two  figures  each, 
placing  a  dot  over  units^  place^  another  over  hundreds,  etc. 
The  root  will  have  as  many  figures  as  there  are  periods. 

Remark. — Since  the  square  of  a  number  consisting  of  tens  and 
units,  is  equal  to  the  square  of  the  tens,  etc.,  when  a  number  has  two 
periods,  it  follows  that  the  left  hand  period  must  contain  the  squart 
of  the  tens  01  first  figure  of  the  root.    (Art.  525.) 


EXTRACTION     OF    THE     SQUARE    ROOT. 
535.  To  extract  the  Square  Hoot  of  a  Number. 

I.  A  man  wishes  to  lay  out  a  garden  in  the  form  of  a 
square,  which  shall  contain  625  sq.  yards:  what  will  be 
the  length  of  one  side  ? 

Analysis. — Since  625  contains  two  periods,  its  root       operation. 
will  have  two  figures,  and  the  left  hand  period  con-  r    '(    ^ 

tains  the  sgw^re  of  the  tens' figure.     (Art.  534,  ^^m.)  ^ 

But  the  greatest  square  of  6  is  4,  and  the  root  of  4  ^ 

is  2,  which  we  place  on  the  right  for  the  tens'  figure      45)2  25 
of  the  root.     Now  the  square  of  2  tens  or  20  is  400,  225 

and  625—400=225.    Hence,  225  is  twice  the  product  of 
the  tens'  figure  of  the  roct  into  the  units,  plus  the  square  of  the 
units.     (Art.  525.) 


534.  How  many  figures  has  the  square  root  of  a  nnmLer  ? 


336 


EXTRACTIOl?^     OF     SQUARE     ROOT. 


20  yds. 

syds. 

m 

T! 

>. 

w^.:^.^..^ 

fO 

-  ^ 

O 

^^^:;=^"— ==tr}f-g 

yds. 


syds. 


But  one  of  tlie  factors  of  this  product  is  2  times  20  or  40 ;  there* 
fore  the  other  factor  must  be  225  divided  by  40 ;  and  «25-f-40=5. 
the  factor  required.  (Art.  93.)  Taking  2  times  20  into  5=200 
(i,  e.,  twice  the  product  of  the  tens  into  the  units'  figure  of  the  root) 
from  225,  leaves  25  for  the  square  of  the  units'  figure,  the  square 
root  of  which  is  5  Hence  25  is  the  square  root  of  625,  and  is  there- 
fore the  length  of  one  side  of  the  garden. 

2d  Analysis. — Let  the  shaded  part  of 
the  diagram  be  the  square  of  2  tens,  the 
first  figure  of  the  root ;  then  20  x  20,  or 
400  sq.  yds.,  will  be  its  contents.  Sub- 
tracting the  contents  from  the  given  area, 
we  have  625—400=225  sq.  yds.  to  be  added 
to  it.  To  preserve  its  form,  the  addition 
must  be  made  equally  to  two  adjacent 
sides.  The  question  now  is,  what  is  the 
width  of  the  addition. 

Since  the  length  of  the  plot  is  20  yds.,  adding  a  strip  i  yard  wide 
to  two  sides  will  take  20  +  20  or  40  sq.  yds.  Now  if  40  sq.  yds.  will 
add  a  strip  i  yard  wide  to  the  plot,  225  sq.  yds.  will  add  a  strip  as 
many  yds.  wide  as  40  is  contained  times  in  225  ;  and  40  is  contained 
in  225,  5  times  and  25  over. 

That  is,  since  the  addition  is  to  be  made  on  two  sides,  we  double 
the  root  or  length  of  one  side  for  a  trial  divisor,  and  find  it  is  con- 
tained in  225,  5  times,  which  shows  the  width  of  the  addition  to  be 
5  yards. 

Now  the  length  of  each  side  addition  being  20  yds.,  and  the  width 
5  yds.,  the  area  of  both  equals  20  x  5  +  20  x  5,  or  40  x  5  =  200  sq.  yards. 
But  there  is  a  vacancy  at  the  upper  corner  on  the  right,  whose  length 
and  breadth  are  5  yds.  each  ;  hence  its  area =5  x  5,  or  25  sq.  yards; 
and  200  sq.  yd.  +  25  sq.  yd.  =  225  sq.  yd.  For  the  sake  of  finding  the 
area  of  the  two  side  additions  and  that  of  the  corner  at  the  same 
time,  we  place  the  quotient  5  on  the  right  of  the  root  already  found, 
and  also  on  the  right  of  the  trial  divisor  to  complete  it.  Multiplying 
the  divisor  thus  completed  by  5,  the  figure  last  placed  in  the  root, 
we  have  45x5  =  225  sq.  yds.  Subtracting  this  product  from  the 
tfiividend,  nothing  remains.     Therefore,  etc.     Hence,  the 


535.  What  is  the  first  step  in  extracting  the  square  root  of  a  number  ?  The 
eecond?  Third?  Fourth?  How  proved?  iVofe.  If  the  trial  divisor  is  not  con- 
tained in  the  dividend,  what  must  be  done  ?  If  there  is  a  remainder  after  the 
root  of  the  last  period  is  found,  what?  How  many  decimals  does  the  root  of  a 
decimal  fraction  have  I 


EXTRACTION    OF    SQUARE    ROOT.  337 

EuLE. — I.  Divide  the  number  i7ito  periods  of  two  figures 
each,  putting  a  dot  over  units,  then  over  every  second  figure 
towards  the  left  in  whole  numbers,  and  towards  the  right 
in  decimals, 

II.  Find  the  greatest  square  in  the  left  hand  period,  and 
place  its  root  on  the  right.  Subtract  this  square  from  the 
period,  and  to  the  right  of  the  remainder  bring  down  the 
next  period  for  a  dividend, 

III.  Double  the  part  of  the  root  thus  found  for  a  trial 
divisor;  and  finding  how  many  times  it  is  contained  in  the 
dividend,  excepting  the  right  hand  figure,,  amiex  the  quotient 
both  to  the  root  and  to  the  divisor, 

IV.  Multiply  the  divisor  thus  increased  by  this  last 
figure  placed  in  the  root,  subtract  the  product,  and  bring 
down  the  next  period. 

V.  Double  the  right  hand  figure  of  the  last  divisor,  and 
proceed  as  before,  till  the  root  of  all  the  periods  is  found. 

Proof. — Multiply  the  root  into  itself,    (Art  528.) 

Notes. — i.  If  the  trial  divisor  is  not  contained  in  the  dividend, 
annex  a  cipher  both  to  the  root  and  to  the  divisor,  and  bring  down 
the  next  period. 

2.  Since  the  product  of  the  trial  divisor  into  the  quotient  figuro 
cannot  exceed  the  dividend,  allowance  must  be  made  for  carrying,  if 
the  product  of  this  figure  into  itself  exceeds  9. 

3.  It  sometimes  happens  that  the  remainder  is  larger  than  the 
divisor ;  but  it  does  not  necessarily  follow  from  this  that  the  figure 
in  the  root  is  too  small,  as  in  simple  division. 

4.  If  there  is  a  remainder  after  the  root  of  the  last  period  is  found, 
annex  periods  of  ciphers,  and  proceed  as  before.  The  figures  of  the 
root  thus  obtained  will  be  decimals. 

5.  The  square  root  of  a  decimal  fraction  is  found  in  the  same  way 
as  that  of  a  whole  number ;  and  the  root  will  have  as  many  decimal 
figures  as  there  are  periods  of  decim^ils  in  the  given  number. 

6.  The  left  hand  period  in  whole  numbers  may  have  but  one  figure  ; 
but  in  decimals,  each  period  must  have  two  figures.  (Art.  533.) 
Hence,  if  the  number  has  but  one  decimal  figure,  or  an  odd  number 
of  decimals,  a  cipher  must  be  annexed  to  complete  the  period. 

536.   Reasons. — i.  Dividing  the  number  into  periods  of  two 
figures  each,  shows  how  many  figures  the  root  will  contain,  and 
15 


338  EXTRACTIOi^     OF     SQUAllE     KOOT. 

enables  us  to  find  its  first  figure.  For,  the  left  hand  period  containa 
the  square  of  this  figure,  and  from  the  square  the  root  is  easily 
found.     (Art.  534,  Rem.) 

'1.  Subtracting  the  square  of  the  first  figure  of  the  root  from  the 
left  hand  period,  shows  what  is  left  for  the  other  figures  of  the  root. 

3.  The  object  of  doubling  the  first  figure  of  the  root,  and  dividing 
the  remainder  by  it  as  a  trial  divisor,  is  to  find  the  next  figure  of  tho 
root.  The  remainder  contains  twice  the  product  of  the  teiis  into  the 
units;  consequently,  dividing  this  product  by  double  the  tens'  factor, 
the  quotient  will  be  the  other  factor  or  units'  figure  of  the  root. 

4.  Or,  referring  to  the  diagram,  it  is  doubled  because  the  remainder 
must  be  added  to  two  sides,  to  preserve  the  form  of  the  square. 

5.  The  right  hand  figure  of  the  dividend  is  excepted,  to  counter- 
balance the  omission  of  the  cipher,  which  properly  belongs  on  the 
right  of  the  trial  divisor. 

6.  The  quotient  figure  is  placed  in  the  root;  it  is  also  annexed  to 
the  trial  divisor  to  complete  it.  The  divisor  thus  completed  is 
multiplied  l)y  the  second  figure  of  the  root  to  find  the  contents  of  the 
additions  thus  made. 

The  reasons  for  the  steps  in  obtaining  other  figures  of  the  root 
mi^  be  shown  in  a  similar  manner. 

2.  What  is  the  square  root  of  381.0304?         Ans.  19.52. 
Suggestion. — Since  the  number  contains  decimals,  we  begin  at 

the  units'  place,  and  counting  both  ways,  have  four  periods ;  as, 
381.0304.  The  root  will  therefore  have  4  figures.  But  there  are  two 
periods  of  decimals;  hence  we  point  off  two  decimals  in  the  root. 

3.  What  is  the  square  root  of  1 01 2036  ?  Ans.  1006. 

4.  What  is  the  square  root  of  2  ?  A7is.  1.4 142 1  +. 

Extract  the  square  root  of  the  following  numbers : 


5. 182329. 

II.  .1681. 

17.5. 

23.  19.5364. 

6.  516961. 

12.  .725. 

18.7. 

24.  3283.29. 

7-  595984. 

13.  .1261. 

19.  8. 

25.  87.65. 

8.  3.580. 

14.  2.6752. 

20.  10. 

26.  123456789. 

9.  .4096. 

15.  4826.75. 

21.  II. 

27.  61723020.96. 

10.  .120409. 

16.  452.634. 

22.  12. 

28.  9754.60423716. 

536.  Why  divide  the  number  into  periods  of  two  figures  each  ?  Why  subtract 
the  square  of  the  first  figure  from  the  period  ?  Why  double  the  first  figure  of  the 
root  for  a  trial  divisor  ?  Why  omit  the  right  hand  figure  of  the  dividend  ?  Why 
place  the  quotient  figure  on  the  right  of  the  trial  divisor?  Why  multiply  il.e 
trial  divisor  thus  completed  by  the  figure  last  placed  in  the  root? 


APPLICATION'S    OF    SQUARE    ROOT.         339 

637.  To  find  the  Square  Root  of  a  Common  Fraction. 

I.  When  the  numerator  and  denominator  are  both  perfect 
squares,  or  can  he  reduced  to  such,  extract  the  square  root 
of  each  term  separately. 

II.  When  they  are  imperfect  squares,  reduce  them  to 
decimals,  and  proceed  as  above. 

Note. — To  find  the  square  root  of  a  mixed  nuiriber,  reduce  it  to 
an  improper  fraction,  and  proceed  as  befora 

29.  What  is  the  square  root  of  y^g^? 
Analysis.— 1%=^,  and  t/^=|,  Arts. 

30.  What  is  the  square  root  off?  Ans..'j'j4S  +' 

31.  What  is  the  square  root  of  if?  Ans.  1.1726  +. 

rind  the  square  root  of  the  following  fractions : 

32.  Hi'  35.  6i.  38.  Iff.  41.  i^V 

•    33'  f  36.  i3i-  39-  i-U'  42.  27^^. 

34.  iV  37.  lyf-  40.  it-f  43-  5ifi- 

APPLICATIONS. 

538.    To  find  the  Side  of  a   Square  equal  in  area  to  a 
given  Surface. 

1.  Find  the  side  of  a  square  farm  containing  40  acres. 
Analysis. — In  i  acre  there  are  160 

1  J      .  ,.  OPERATION. 

sq.   rods,   and  m  40  acres,   40  times  k  ^        r 

160,  or  6400  sq.  rods.     The   ^^6400=  40  A.  X  160  =  6400  sq.  r. 

80  r.    Therefore  the  side  of  the  farm  is  1/6400  =  80  r.  Ans. 
80  linear  rods.     Hence,  the 

Rule. — I.  Extract  the  square  root  of  the  given  surface. 

Note. — The  root  is  in  linear  units  of  the  same  name  as  the  given 
surface. 

2.  What  is  the  side  of  a  square  tract  of  land  containing 
II 02  acres  80  sq.  rods ? 

537.  How  find  the  square  root  of  a  common  fraction?    Note.  Of  a  mixed 
number?    538.  How  find  the  side  of  a  square  equal  to  a  given  surface  ? 


340 


APPLICATIOif  S    or    SQUARE    ROOT. 


3.  How  many  rods  of  fencing  does  it  require  to  inclose 
a  square  farm  which  contains  122  acres  30  sq.  rods  ? 

4.  A  bought  141 6 1  fruit  trees,  which  he  planted  so  r:s 
to  form  a  square :  how  many  trees  did  he  put  in  a  row  ? 

5.  A  general  has  an  army  of  56644  men:  how  many 
must  he  place  in  rank  and  file  to  form  them  into  a  square  ? 

539.   A  Triangle  is  a  figure  "having  three  C 

ddes  and  three  angles. 

A  Hi  (flit-angled  Triangle  is  one  that 
contains  a  right  angle.     (Art.  260.) 

The  side  opposite  tlie  right  angle  is  called  tlio 
hypothenuse ;  the  other  two  sides  the  hose  and 
perpendicular. 

640.  The  square  described  on  the  hypothenuse  of  a  right- 
angled  triangle  is  equal  to  the  sum  of  the  squares  descriied 
on  the  other  two  sides.^ 


541.  The  truth  of  this  principle  may 
be  illustrated  thus :  Take  any  right-angled 
triangle  ABC;  let  the  hypothenuse  h, 
be  5  in.,  the  base  h,  4  in.,  and  the  per- 
pendicular p,  3  in.  it  will  be  seen  that  the 
square  of  h  contains  25  sq.  in.,  the  square 
of  h  16  sq.  in.,  and  the  square  of  ^  9  sq.  in. 
Now  25  =  16  +  9,  which  accords  with  the 
proposition.  In  like  manner  it  may  be 
shown  that  the  principle  is  true  of  all 
right-angled  triangles.     Hence, 


542.  To  find  the  IIyj)ot7ienuse,  the  Base  and  Perpendicular 
being  given. 

To  the  square  of  the  base  add  the  square  of  the  per- 
pendicular, and  extract  the  square  root  of  their  sum. 

539.  What  is  a  triangle?  A  right-angled  triangle?  Draw  a  right-angled 
triangle?  The  side  opposite  the  right-angle  called?  The  other  two  sides? 
540.  To  what  is  the  square  of  the  hypothenuse  equal?  541.  Illustrate  this  prin- 
ciple by  a  figure?  542.  How  find  the  hypothenuse  M-hen  the  base  and  per- 
pendicular are  given  ? 


\A  ) 

n 

m 

1=1 

^mm^ 

B 

- 

- 

*  Thomson's  Geometry,  IV,  1 1 ;  Euclid,  I,  47. 


/- 


applicatio:n^s   of   squabe   boot.       341 

643.  To  find  the  Base,  the  Hypothenuse  and   Perpendicular 
being  given. 

From  the  square  of  the  hypothenuse  take  the  square 
of  the  perpendicular,  and  extract  the  square  root  of  the 
remainder. 

544.    To   find    the  Perpendicular,  the   Hypothenuse    and 
Base  being  given. 

From  the  square  of  the  hypothenuse  take  the  square 
of  the  hase,  and  extract  the  square  root  of  the  remainder. 

Note. — The  pupil  should  draw  figures  corresponding  with  the 
conditions  of  the  following  problems,  and  indicate  the  parts  given : 

6.  The  perpendicular  height  of  a  flag-staff  is  ^6  ft.: 
what  length  of  Hne  is  required  to  reach  from  its  top  to  a 
point  in  a  level  surface  48  ft.  from  its  base  ? 

SOLUTlOlf.— The  square  of  the  base    .  .  =48  x  48=2304 
'*        "        perpendicu]ar=36  X  36  =  1296 
The  square  root  of  their  sum =1/3600 =60  ft.  Ans. 

7.  The  h}"pothenuse  of  a  right-angled  triangle  is  135  yds., 
the  perpendicular  81  yds.:  what  is  the  base? 

8.  One  side  of  a  rectangular  field  is  40  rods,  and  the 
distance  between  its  opposite  corners  50  rods:  what  is  the 
length  of  the  other  side  ? 

9.  Two  vessels  sail  from  the  same  point,  one  going 
due  south  360  miles,  the  other  due  cast  250  miles:  how 
far  apart  were  they  then  ? 

10.  The  height  of  a  tree  on  the  bank  of  a  river  is  100  ffc., 
and  a  line  stretching  from  its  top  to  the  opposite  side  is 
144  ft. :  what  is  the  width  of  the  river  ? 

11.  The  side  of  a  square  room  is  40  feet:  what  is  the 
distance  between  its  opposite  corners  on  the  floor  ? 

12.  A  tree  was  broken  35  feet  from  its  root,  and  struck 
the  ground  21  ft.  from  its  base:  what  was  the  height  of 
the  tree  ? 

543.  How  find  the  base  when  the  other  two  sides  are  given  ?  544.  How  find 
the  perpendicular  when  the  other  two  sides  are  given  ? 


342  SIMILAR     FIGURES. 

SIMILAR     FIGURES. 

545.  Similar  Figures  are  those  which  have  the 
same  form,  and  their  like  dimensions  proportional. 

Notes. — i.  All  circles,  of  wliatever  magnitude,  are  similar. 

2.  All  squares,  equilateral  triangles,  and  regular  polygons  are 
similar.     And,  universally, 

All  rectilinear  figures  are  similar,  wlien  their  several  angles  are 
equal  each  to  each,  and  their  like  dimensions  proportional. 

3.  The  like  dimensions  of  circles  are  their  diameters,  radii,  and 
circumferences. 

546.  The  areas  of  similar  figures  are  to  each  other  as 
the  squares  of  their  like  diraensions.     And, 

Conversely,  the  liJce  dimensions  of  similar  figures  are  to 
each  other  as  the  square  roots  of  their  areas. 

13.  If  one  side  of  a  triangle  is  12  rods,  and  its  area 
72  sq.  rods,  what  is  the  area  of  a  similar  triangle,  the 
corresponding  side  of  which  is  8  rods  ? 

Solution. — (12)2 :  (8)2 : :  72  :  Ans.,  or  32  sq.  rods. 

14.  If  one  side  of  a  triangle  containing  s^  sq.  rods  is 
8  rods,  what  is  the  length  of  a  corresponding  side  of  a 
similar  triangle  which  contains  81  sq.  rods? 

So:.UTiON. — v^36  :  |/8i  : :  8  :  Ans.,  or  12  rods. 

15.  If  a  pipe  2  inches  in  diameter  will  fill  a  cistern  in 
42  min.,  in  what  time  will  a  pipe  7  in.  in  diameter  fill  it  ? 

547.  To  find  a  Mean  Proportional  between  two  Numbers. 

16.  What  is  the  mean  proportional  between  4  and  16  ? 
Analysis. — When  three  numbers  are  proportional,  the  product  of 

the  extremes  is  equal  to  the  square  of  the  mean.    (Arts.  487,  490.) 
But  4  X  16=64  ;  and  1/64=8.  Ans.     Hence,  the 

EuLE. — Extract  the  square  root  of  their  product. 

Find  the  mean  proportional  between  the  following : 
17.  4  and  36.  19.  56  and  72.  21.  Jf  and  -^. 

t8.  36  and  81.  20.  .49  and  6.25.  22.  -^  and  ^^V 

545.  What  are  similar  figures  ?    546.  How  are  tkeir  areas  to  each  other  V 


FORMATIOK     OF     CUBES. 


543 


FORMATION     OF     CUBES. 
548.  To  find  the  Cube  of  a  Number  In  the  terms  of  its  parts. 

I.  FiaJ  the  cube  of  32  in  the  terms  of  the  parts  30  and  2. 

ANALYSl!^.— 32=30  4-2,  and  325  =  (30  +  2)x(30  4-2)x(30  +  2.) 


96    : 

64: 


:3^_+2 

•3^^ +  (30x2) 

:  (30  X  2)  4-  2^ 

:302  +  2Xv30 


1024 

32  =  30  +2 


2)  +  22 


3072    =30^  + 2  X  (30'' X  2) +-(30x2*^) 
2048  —  (30-  X  2)  +  2  X  v30  X  2'^)  +  28 


32768  =  30^  +  3  X  (3o''^  X  2)  +  3  X  ,;30  X  z^)  +  2- 


3  feet. 


3X.3X3 


Or  thus :  Let  the  diagram  represent  a 
cube  whose  side  is  30  ft. ;  its  contents= 
tlie  cube  of  3  tens,  or  2700  cu.  ft. 

What  additions  must  be  made  to  this 
cube,  and  hoic  made,  to  preserve  its/orw, 
and  make  it  equal  to  the  cube  of  32. 

ist.  To  preserve  its  cubical  form,  the 
additions  must  be  equally  made  on  tliree 
adjacent  sides ;  as  the  top,  front,  and  right. 

2d.  Since  32  is  2  more  than  30,  it  follows  that  this  cube  must  be 
made  2  ft.  longer,  2  ft.  wider,  and  2  ft.  higher,  that  its  lengt^, 
breadth,  and  thickness  may  each  be  32  ft. 

But  as  the  side  of  this  cube  is  30  ft.,  the  contents  of  each  of  these 
additions  must  be  equal  to  the  square  of  the  tens  (so^)  into  2,  the  units, 
and  their  sum  must  be  3  times  (30*  x  2)=3  x  gcx)  x  2=5400  cu.  ft. 

But  there  are  three  vacancies  along  the  edges  of  the  cube  adjacent 
to  the  additions.  Each  of  these  vacancies  is  30  ft.  long,  2  ft.  wide, 
and  2  ft.  tliick  ;  hence,  the  contents  of  each  equals  30  x  2*,  and  the  sum 
of  their  contents  equals  3  times  the  tens  into  the  square  of  the  units 
=  3  times  (30  X  2^)= 3  X  30  X  4=360  cu.  ft.  But  there  is  still  another 
vacancy  at  the  junction  of  the  corner  additions,  whose  length, 
breadth,  and  thickness  are  each  2  ft.,  and  whose  contents  are  equal 
to  2^=2x2x2=8.  The  cube  is  now  complete.  Therefore  32^= 
27000  (30^) +  5400  (3  times  30^x2) +  360  (3  times  30x2^)4-8  (2^)  = 
32768  cu.  ft.     In  like  manner  it  may  be  shown  that. 


548.  To  what  IB  the  cube  of  a  number  consisting  of  tens  and  units  equal 


344  FORMATION     OF     CUBES. 

TJie  cube  of  any  number  consisting  of  tens  and  units  is 
equal  to  the  cube  of  the  tens^  plus  3  times  the  square  of  the 
tens  into  the  miits^  plus  3  times  the  tefis  into  the  square  of 
the  units,  plus  the  cube  of  the  units. 

549.  To  find  how  many  figures  the  Cube  of  a  Number  contains. 

ist.  Take  i  and  g,  also  10  and  99,  100  and  999,  etc.,  the  least  and 
greatest  int^ers  that  can  be  expressed  by  one,  two,  three,  etc.,  figures. 

2d.  In  like  manner  talie  .1  and  .9,  also  .01  and  .99,  etc.,  the  least 
and  greatest  decimals  that  can  be  expressed  by  one,  two,  etc.,  decimal 
figures.    Cubing  these,  we  have 


t  I  and  1^=1,  -  -  - 

.1  and 

.l3=.I 

9  "   9^-729,   -  - 

-   -9  " 

.93  =  . 729 

10  "   10^=1000,  -  - 

-  .01  " 

.01 3  =.000001 

99  "   993=970299,   - 

-  -99  " 

.993=. 970299 

100  "  1003=1000000,  - 

-  .001  " 

.ooi^  .000000001 

999  "  999^=997002999, 

-  .999  " 

.9993=. 997002999 

By  comparing  these  roots  and  their  cubes,  we  discover  that 
T/ie  cube  of  a  number  canfiot  have  more  than  three  times 
as  many  figures  as  its  root,  nor  but  tivo  less.    Hence, 

550.  To  find  how  many  figures  the  Cube  Soot  contains. 

Divide  the  number  into  periods  of  three  figures  each, 
putting  a  dot  over  units,  then  over  every  third  figure 
towards  the  left  in  whole  numbers,  and  towards  the  right 
in  decimals. 

Remakes. — i.  Since  the  cube  of  a  number  consisting  of  tens  and 
units  is  equal  to  the  cvhe  of  the  tens,  plus  3  times  the  square  of  the 
tens  into  the  units,  etc.,  when  a  number  has  two  periods,  it  follows 
that  the  left  hand  period  must  contain  the  cube  of  the  tenSy  or  first 
figure  of  the  root. 

2.  The  left  hand  period  in  whole  numbers  may  be  incomplete, 
having  only  wie  or  two  figures;  but  in  decimals  each  period  must 
always  have  Viree  figures.  Hence,  if  the  decimal  figures  in  a  given 
number  are  less  than  three,  annex  ciphers  to  complete  the  period. 

How  many  figures  in  the  cube  root  of  the  following : 

2.  340566.  4.  576.453.  6.  32.7561. 

3.  1467.  5.  5.7321.  7.  .456785. 


549.  How  many  figures  has  the  cube  of  a  number? 


EXTEACTIOI^     OF    THE     CUBE     ROOT.        345 

EXTRACTION  OF  THE  CUBE  ROOT. 

551.  To  extract  the  Cube  Root  of  a  Number. 

I.  A  man  having  32768  marble  blocks,  each  being  a 
cubic  foot,  wishes  to  arrange  them  into  a  single  cube: 
what  must  be  its  side  ? 

Analysis. — Since   32768  contains  two   periods,  operation. 

we  know  its  root  will  have  two  figures;  also  that  ■^2768(-^2 

the  left  hand  period  contains  the  cube  of  the  tens  27 

or  first  figure  of  the  root.     (Art.  550,  Rem)  ■ 

The  greatest  cube  in  32  is  27,  and  its  root  is  3,  '  °° 

which  we  place  on  the  right  for  the  tens  or  first  ^ 
figure  of  the  root.     Subtracting  its  cube  from  the 


5768 
5768 


first  period,  and  bringing  down  the  next  period  2884 
to  the  right  of  the  remainder,  we  have  5768,  which 
by  the  formation  of  the  cube  is  equal  to  3  times  the  square  of  the 
tens'  figure  into  the  units,  plus  3  times  the  tens  into  the  square  of 
the  units,  plus  the  cube  of  the  units.  We  therefore  place  3  times 
the  square  of  the  tens  or  2700,  on  the  left  of  the  dividend  for  a 
trial  divisor ;  and  dividing,  place  the  quotient  2  on  the  right  for  the 
units'  figure  of  the  root. 

To  complete  the  divisor,  we  add  to  it  3  times  30  the  tens  into  2 
units=i8o;  also  4,  the  square  of  the  units,  making  it  2884.  Multi- 
plying the  divisor  thus  completed  by  the  units'  figure  2,  we  have 
2884  X  2  =  5768,  the  same  as  the  dividend.     Ans.  32  ft. 

Or  thus :  Let  the  cube  of  30,  the  tens  of  the  root,  be  represented 
by  the  large  cube  in  the  set  of  cubical  blocks.*  The  remainder  5768, 
is  to  be  added  equally  to  three  adjacent  sides  of  this  cube. 

To  ascertain  the  thickness  of  these  side  additions,  we  form  a  trial 
divisor  by  squaring  3,  the  first  figure  of  the  root,  with  a  cipher 
annexed,  for  the  area  of  one  side  of  this  cube,  and  multiply  this 
square  by  3  for  the  three  side  additions.  Now  30^=30  x  30=900; 
and  900x3  =  2700,  the  trial  divisor.  Dividing  5768  by  2700,  the 
quotient  is  2,  which  shows  that  the  side  additions  are  to  be  2  ft. 
thick,  and  is  placed  on  the  right  for  the  units'  figure  of  the  root. 

551.  The  first  step  in  extracting  the  cube  root  f  The  second?  Third? 
Fourth?  Fifth?  Note.  If  the  trial  divisor  is  not  contained  in  the  dividend,  how 
proceed  ?    If  there  is  a  remainder  after  the  root  of  the  last  period  is  found,  how  f 


*  Every  school  in  which  the  cube  root  is  taught  is  presumed  to  be  furnished 
with  a  set  of  Cubical  Blocks.  '' 


S46        EXTRA  CTIOK     OF    THE      CUBE     EOOT. 

To  represont  these  additions,  place  the  corresponding  layers  on 
the  top,  front,  and  right  of  the  large  cube.  But  we  discover  three 
vacancies  along  the  edges  of  the  large  cube,  each  of  which  is  30  ft. 
long,  2  ft.  wide,  and  2  ft.  thick.  Filling  these  vacancies  with  the 
corresponding  rectangular  blocks,  we  discover  another  vacancy  at 
the  junction  of  the  corners  just  filled,  whose  length,  breadth,  and 
thickness  are  each  2  ft.     This  is  filled  by  the  small  cube. 

To  complete  the  trial  divisor,  wo  add  the  area  of  one  side  of  each 
of  the  corner  additions,  viz.,  30  x  2  x  3,  or  180  sq.  ft.,  also  the  area  of 
one  side  of  the  small  cube=2  x  2,  or  4  sq.  ft.  Now  2700+  180  +  4= 
2884.  The  divisor  is  now  composed  of  the  area  of  3  sides  of  the 
large  cube,  plus  the  area  of  one  side  of  each  of  the  corner  additions, 
plus  the  area  of  one  side  of  the  small  cube,  and  is  complete. 

Finally,  to  ascertain  the  contents  of  the  several  additions,  we 
multiply  the  divisor  thus  completed  by  2,  the  last  figure  of  the  root  ; 
and  2884x2  =  5768.  (Art.  249.)  Subtracting  the  product  from  the 
dividend,  nothing  remains.     Hence,  the 

Rule. — I.  Divide  the  number  into  periods  of  three  figures 
each,  putting  a  dot  over  units,  then  over  every  third  figure 
towards  the  left  in  ivhole  numbers,  and  towards  the  right 
in  decimals. 

II.  Find  the  greatest  cube  in  the  left  hand  period,  and 
place  its  root  on  the  right.  Subtract  its  cube  from  the 
period,  and  to  the  right  of  the  remainder  bring  down  the 
next  period  for  a  dividend. 

III.  Multiply  the  square  of  the  root  thus  found  ivith  a 
cipher  annexed,  by  three,  for  a  trial  divisor ;  and  finding 
how  many  times  it  is  contained  in  the  dividend,  write  the 
quotient  for  the  second  figure  of  the  root. 

IV.  To  complete  the  trial  divisor,  add  to  it  three  times 
the  product  of  the  root  'previously  found  with  a  cipher 
annexed,  into  the  second  root  figure,  also  the  square  of  the 
second  root  figure. 

V.  Multiply  the  divisor  thus  completed  by  the  last  figure 
placed  in  the  root  Subtract  the  product  from  the  divi- 
dend ;  and  to  the  right  of  the  remainder  bring  down  the 
next  period  for  a  neio  dividend.  Find  a  neio  trial  divisor, 
as  before,  and  th^  proceed  till  the  root  of  the  last  period  is 
found. 


EXTRACTIOi^     OF    THE     CUBE     ROOT.        347 

Notes. — i.  If  a  trial  divisor  is  not  contained  in  tlie  dividend,  put 
a  cipher  in  tlie  root,  tivo  ciphers  on  the  right  of  the  divisor,  and  bring 
down  the  next  period. 

2.  If  the  product  of  tho  divisor  completed  into  the  figure  last 
placed  in  the  root  exceeds  the  dividend,  the  root  figure  is  too  large. 
►Sometimes  the  remainder  is  larger  than  the  divisor  completed ;  but 
it  does  not  necessarily  follow  that  the  root  figure  is  too  small. 

3.  When  there  are  three  or  more  periods  in  the  given  number,  the 
first,  second,  and  subsequent  trial  divisors  are  found  in  the  same 
manner  as  when  there  are  only  two.  That  is,  disregarding  its  true 
local  value,  we  simply  multiply  the  square  of  the  root  already  found 
with  a  cipher  annexed,  by  3,  etc, 

4.  If  there  ip.  a  remainder  after  the  root  of  the  last  period  is  found, 
annex  periods  of  ciphers,  and  pioceed  as  before.  The  root  figures 
thus  obtained  will  bo  decimals. 

5.  The  cube  root  of  a  decimal  fraction  is  found  in  the  same  way  as 
that  of  a  whole  number ;  and  will  have  as  many  decimal  figures  as 
there  are  periods  of  decimals  in  tho  nutuber.     (Art.  549.) 

552.  Reasons. — i.  Dividing  the  number  into  periods  shows 
how  many  figures  the  root  contains,  and  enables  us  to  find  the  firnt 
figure  of  the  root.  For,  the  left  hand  period  contains  the  cube  of 
the  first  figure  of  the  root.     (Art,  550.) 

2.  The  object  of  the  trial  divisor  is  to  find  the  next  figure  of  the 
root,  or  the  thickness  of  the  side  additions.  The  root  is  squared  to 
find  the  area  of  one  side  of  the  cube  whose  root  is  found,  the  cipher 
being  annexed  because  the  first  figure  is  tens.  This  square  is  multi- 
plied by  3,  because  the  additions  are  to  be  made  to  three  sides. 

3.  The  root  previously  found  is  multiplied  by  this  second  figure 
10  find  the  area  of  a  side  of  one  of  the  vacancies  along  the  edges  of 
the  cube  already  found.  This  product  is  multiplied  by  3,  because 
there  are  three  of  these  vacancies ;  and  the  product  is  placed  under 
the  trial  divisor  as  a  correction.  The  object  of  squaring  the  second 
figure  of  the  root  is  to  find  the  area  of  one  side  of  the  cubical  vacancy 
at  the  junction  of  the  corner  vacancies,  and  with  the  other  correction 
this  is  added  to  the  trial  divisor  to  complete  it. 

4.  The  divisor  thus  completed  is  multiplied  by  the  second  figuro 
of  the  root  to  find  the  contents  of  the  several  additions  now  made. 


552.  Why  divide  the  number  into  periods  of  three  fl^irep  ?  "What  is  the  ohject 
of  a  trial  divisor  ?  Why  square  the  root  already  found  ?  Why  annex  a  cipher  to 
it  ?  Why  multiply  this  square  by  3  ?  Why  is  the  root  previously  found  multi- 
plied by  the  second  figure  of  the  root?  Why  multiply  this  product  by  3?  Why 
square  tbe  second  fignre  of  the  root  ?  Why  multiply  the  divisor  rrhen  completed 
by  the  second  flg:ure  of  the  root  * 


348        EXTRACTION     OF    THE     CUBE     ROOT. 

2.  What  is  the  cube  root  of  1 30241.3  ? 

Anaxysis. — Having  completed  the  operatiok. 

period  of  decimals  by  annexing  two  i-:jo24I.':joo(c;o  6  + 

ciphers,  we  find  the  first  figure  of  the  j  2  c 


root  as  above.     Bringing  down  the 
next    period,   the    dividend    is    5241.      '^ 
The  trial  divisor  7500  is  not  contained  " 

'in  the  dividend;  therefore  placing  a  ^ 


5241.300 
4554216 


cipher  in  the   root   and  two  on  the     759*^3^ 

right  of  the  divisor,  we  bring  down  687084  Rem. 

the  next  period,  and  proceed  as  before. 

3.  Cube  root  of  6 141 25  ?  6.  Cube  root  of  3  ? 

4.  Cube  root  of  84.604  ?  7.  Cube  root  of  21.024576  ? 

5.  Cube  root  of  373248  ?  8.  Cube  root  of  17  ? 

9.  What  is  the  cube  root  of  705919947264? 

10.  What  is  the  cube  root  of  .253395799  ? 

11.  What  is  the  side  of  a  cube  which  contains  628568 
cu.  yards  ? 

Note. — The  root  is  in  linear  units  of  the  same  name  as  the  given 
contents. 

12.  What  is  the  side  of  a  cube  equal  to  a  pile  of  wood 
40  ft.  long,  15  ft.  wide,  and  6  ft.  high  ? 

13.  What  is  the  side  of  a  cubical  bin  which  will  hold 
1000  bu.  of  corn;  allowing  2150.4  cu.  in.  to  a  bushel  ? 

553.  To  find  the  Cube  Root  of  a  Common  Fraction, 

If  the  numerator  and  denominator  are  perfect  cules,  of 
can  he  reduced  to  such,  extract  the  cube  root  of  each. 

Or,  reduce  the  fraction  to  a  decimal,  and  proceed  as  before* 
Note. — ^Reduce  mixed  numbers  io  improper  fractions,  etc. 

14.  What  is  the  cube  root  of  /^  ?  .  Ans.  %. 

15.  Cube  root  of  f?  ^?zs.  .75+. 

16.  Cube  root  of  ^^^^  ?  18.  Cube  root  of  ^\^i^  ? 

1 7.  Cube  root  of  \Uil  ?  1 9.  Cube  root  of  8 1 1  ? 

553   How  find  the  cube  root  of  a  common  fraction  ?    Note.  Of  a  mixed  number  f 


SIMILAR     SOLIDS.  349 

APPLICATIONS. 

554.  Similar  Solids  are  those  which  have  the  same 
form,  and  their  like  di7nensions  'projportional. 

Notes. — i.  The  like  dimensions  of  spheres  are  their  diameters, 
radii,  and  circumferences ;  those  of  cubes  are  their  sides. 

2.  The  like  dimensions  of  cylinders  and  cones  are  their  altitudes, 
and  the  diameters  or  the  circumferences  of  their  bases. 

3.  Pyramids  are  similar,  when  their  bases  are  similar  polygons, 
and  their  altitudes  proportional. 

4.  Polyhedrons  (i.  e.,  solids  included  by  any  number  of  plane  faces) 
are  similar,  when  they  are  contained  by  the  same  number  ot  similai 
polygons,  and  all  their  solid  angles  are  equal  each  to  each. 

555.  The  C07itents  of  similar  solids  are  to  each  other  aa 
the  cubes  of  their  lihe  dimensions;  and, 

Conversely,  the  like  dimensions  of  similar  solids  are  a& 
the  cube  roots  of  their  contents. 

1.  If  the  side  of  a  cuhical  cistern  containing  1728  cu.  in. 
is  12  in.,  what  are  the  contents  of  a  similar  cistern  whoso 
side  is  2  ft.  Ans.  i  ^  :  2^  : :  1728  cu.  in. :  con.,  or  13824  cu.  in. 

2.  If  the  side  of  a  certain  mound  containing  74088  cu.  ft. 
is  84  ft.,  what  is  the  side  of  a  similar  mound  which  con- 
tains 17576  cu.  ft.? 

3.  If  a  globe  4  in.  in  diameter  weighs  9  lbs ,  what  is  the 
weight  of  a  globe  8  in.  in  diameter?  A71S.  72  lbs 

4.  If  8  cubic  piles  of  wood,  the  side  of  each  being  8  ft.^ 
were  consolidated  into  one  cubic  pile,  what  would  be  the 
length  of  its  side  .? 

5.  If  a  pyramid  60  ft.  high  contains  12500  cu.  ft.,  what 
are  the  contents  of  a  similar  pyramid  whose  height  is  20  ft.  ? 

6.  If  a  conical  stack  of  hay  15  ft.  high  contains  6  tons, 
what  is  the  weight  of  a  similar  stack  whose  height  is  1 2  ft.  ? 

554.  What  are  similar  solids  ?  Note.  What  are  like  dimen?ion8  of  sphere?  1 
Of  cylinders  and  cones  ?    Of  pyramids  ?    555.  What  relation  have  similar  solids  ? 


AEITHMETICAL    PROGEESSIOK 

556.  An  AintJunetical  I^rogression  is  a  series 
"jf  numbers  which  increase  or  decrease  by  a  common  dif- 
ference. 

Note. — The  numbers  forming  the  series  are  called  the  Terms, 
^he  first  and  last  terras  are  the  Extremes;  the  intermediate  terms 
the  Meav^.    (xirts.  476,  489.) 

557.  The  Cojnmon  Difference  is  the  difference 
between  the  successive  terms. 

558.  ^n  Ascendiufj  Series  is  one  in  which  the  successive 
terms  increase ;  as,  2,  4,  6,  8,  10,  etc.,  the  common  difference  being  2. 

A  Descetiding  Series  is  one  in  which  the  successive  terms 
decrease ;  as,  15,  12,  g,  6,  etc.,  the  common  difference  being  3. 

559.  In  arithmetical  progression  there  are  five  parts  or  elements 
to  be  considered,  viz.:  the  first  term,  the  last  term,  the  number  of 
t:rms,  the  common  difference,  and  the  sum  of  all  the  terms.  Theeo 
parts  are  so  related  to  each  other,  that  if  any  three  of  them  are  given, 
the  other  two  may  be  found. 

560.  To  find  the  Last  Term,  the  First  Term,  the  Number 
of  Terms,  and  the  Common  Difference  being  given. 

I.  Eequired  the  last  term  of  the  ascending  series  having 
7  terms,  its  first  term  being  3,  and  its  common  difference  2. 

Analysis. — From  the  definition,  each  succeeding  term  is  found 
by  adding  the  common  difference  to  the  preceding.     The  series  is : 
3,    3  +  2,    3  +  (2  +  2),    3  +  (2  4-  2  +  2),    3  +  (2  +  2  +  2  +  2),  etc.     Or 
3»    3  +  2,    3  +  (2  X  2),    3  +  (2  X  3),  3  +  (2  X  4),  etc.     That  is, 

561.  The  last  term  is  equal  to  the  first  term,  increased  by 
the  product  of  the  common  difference  into  the  number  of  terms 
less  I.     Hence,  the 

Rule. — Multiply  the  common  difference  hy  the  number 
of  terms  less  i,  and  add  the  product  to  the  first  term. 

556.  What  is  an  arithmetical  progression?  Note.  The  first  and  last  terms 
called?  Tiie  intervening;?  e,^j.  The  common  difference?  558.  An  ascendinsf 
peries?  Descending?  560.  tlow  find  the  last  term,  when  the  first  term,  tlie 
number  of  terms,  and  the  common  difference  are  given? 


ARITHMETICAL     P  R  O  G  R  E  S  S  I  O  iq".  351 

;N'OTES. — I.  In  a  descending  series  the  product  must  be  subtracUd 
from  tlie  first  term  ;  for,  in  this  case  each  succeeding  term  is  found 
by  subtracting  the  common  difference  from  the  preceding  terms, 

2.  Any  term  in  a  series  may  be  found  by  the  i)receding  rule. 
For,  the  series  may  be  supposed  to  stop  at  any  term,  and  that  may 
be  considered,  for  the  time,  as  the  last. 

3.  If  the  last  term  is  given,  and  the  first  term  required,  invert  the 
order  of  the  terms,  and  proceed  as  above. 

2.  If  the  first  term  of  an  ascending  series  is  5,  the 
common  difference  3,  and  the  number  of  terms  1 1,  what 
is  the  last  term  ?  Ans.  2>S- 

3.  The  first  term  of  a  descending  series  is  35,  the  com- 
mon difference  3,  and  the  number  of  terms  10 :  what  is 
the  last  ? 

4.  The  last  term  of  an  ascending  series  is  77,  the  number 
of  terms  19,  and  the  common  difference  3:  what  is  the 
first  term  ?  Ans.  2t^. 

5.  What  is  the  amount  of  $150,  at  7^  simple  interest, 
for  20  years  ? 

562.    To  find  the  Number  of  Tenns,  the  Extremes,  and 
the  Common  Difference  being  given. 

6.  The  extremes  of  an  arithmetical  series  are  4  and  37, 
and  the  common  difference  3 :  what  is  the  number  of  terms  ? 

Analysis. — The  last  term  of  a  series  is  equal  to  the  first  term 
increased  or  diminished  by  the  product  of  the  common  difference 
into  the  number  of  terms  less  1.  (Art.  561.)  Therefore  37—4,  or 
33,  is  the  product  of  the  common  difference  3,  into  the  number  of 
terms  less  i.  Consequently  33-^3  or  11,  must  be  the  number  of  terras 
less  I ;  and  11  +  i,or  12,  is  the  answer  required.   (Art.  93.)    Hence,  the 

Rule. — Divide  the  difference  of  the  extremes  ly  the  com- 
mon difference,  and  add  i  to  the  quotient. 

7.  The  youngest  child  of  a  family  is  i  year,  the  oldest. 
21,  and  the  common  difference  of  their  ages  2  y. :  how 
many  children  in  the  family  ? 

562.  How  find  the  number  of  terms,  when  the  extremes  and  common  difiercnce 
are  given  ? 


352  AKITHMETICAL     PKOGRESSIOK. 

563.  To  find  the  Coinnion  Difference^  the   Extremes  and 

the  Number  of  Terms  being  given. 

8.  The  extremes  of  a  series  are  3  and  21,  and  the  number 
of  terms  is  10 :  what  is  the  common  difference  ? 

Analysis. — The  difference  of  the  extremes  21—3=^18,  is  the  pro- 
duct of  the  number  of  terms  less  i  into  the  common  difference,  and 
10—  I,  or  g,  is  the  number  of  terms  less  i ;  therefore  18-5-9,  or  2,  is  tlie 
common  difiference  required.     (Art.  93.)     Hence,  the 

Rule. — Divide  the  difference  of  the  extremes  ly  the 
number  of  terms  less  i. 

9.  The  ages  of  7  sons  form  an  arithmetical  series,  the 
youngest  being  2,  and  the  eldest  20  years:  what  is  the 
difference  of  their  ages  ? 

564.  To  find  the  Sum  of  all  the  Terms,  the  Extremes  and 

the  Number  of  Terms  being  given. 

10.  Required  the  sum  of  \hQ  series  having  7  terms,  tlie 
extremes  being  3  and  15. 

Analysis — (i.)  The  series  is  3,     5,     7,     g,    11,    13,    15. 

(2.)  Inverting  the  same,     15,    13,    11,      9,      7,      5,      3. 

(3.)  Adding  (i.)  and  (2.),      18  4- 18  +  18  +  18  +  18  +  18  f  i8=twice  the  sum. 

(4.)  Dividing  (3.)  by  2,        9  +  9  +  9  +  9  +  g  +  9  +  9=63,  the  sum. 

By  inspecting  these  series,  we  discover  that  half  the  sum  of  the 
extremes  is  equal  to  the  average  value  of  the  terms.     Hence,  the 

Rule. — Multiply  half  the  sum  of  the  extremes  by  the 

number  of  terms. 

Remark. — From  the  preceding  illustration  we  also  see  that, 
The  sum  of  the  extremes  is  equal  to  the  sum  of  any  two  terms 

equidistant  from  them ;  or,  to  twice  the  sum  of  the  middle  terni ,  if 

the  number  of  terms  he  odd. 

11.  How  many  strokes  does  a  common  clock  strike  in 
12  hours? 


563.  How  find  the  common  difference  when  the  extremes  and  number  of  terma 
are  given  ?  564.  How  find  the  sum  of  all  the  terUiS,  when  the  extremes  twid 
number  of  terms  are  given  ? 


GEOMETEIOAL    PEOGRESSIOK 

565.  A  Geometrical  Progression  is  a  series  of 
numbers  which  increase  or  decrease  by  a  common  ratio. 

Note. — The  series  is  called  Ascending  or  Descending,  according 
as  the  terms  increase  or  decrease.    (Art.  558.) 

566.  In  Geometrical  Progression  there  are  also  five  parts  or 
elements  to  be  considered,  viz.:  the  first  term,  the  last  term,  the 
number  of  terms,  the  ratio,  and  the  sum  of  all  the  terms. 

567.    To  find  the  Last   Term,  the   First  Term,  the  Ratio, 
and  the  Number  of  Terms  being  given. 

1.  Required  the  last  term  of  an  ascending  series  having 
6  terms,  tlie  first  term  being  3,  and  the  ratio  2. 

Analysis. — From  the  definition,  the  series  is 
3*    3x2,    3  X  (2  X  2),    3  X  (2  X  2  X  2),      3  X  (2  X  2  X  2  X  2),  etc.     Or 
3,    3x2,    3  X  2'^,  3x2*^,  3  X  2^,  etc.     That  is. 

Each  successive  term  is  equal  to  the  first  term  multiplied  by  the 
ratio  raised  to  a  power  whoso  index  is  one  less  than  the  number 
of  the  term.     Hence,  the 

Rule. — Multiphj  the  first  term  ly  that  imiver  of  the  ratio 
whose  index  is  i  less  than  the  number  of  terms. 

Notes. — i.  Any  term  in  a  scries  may  be  found  by  the  preceding  rule. 
For,  the  series  may  be  supposed  to  stop  at  that  term. 

2.  If  the  last  term  is  given  and  the  first  required,  invert  the  order 
of  the  terms,  and  proceed  as  above. 

3.  The  preceding  rule  is  applicable  to  Compound  Interest;  the 
principal  being  the  first  term  of  the  series ;  the  amount  of  $1  for  1 
year  the  ratio ;  and  the  number  of  years  plus  i,  the  number  of  terms. 

2.  A  father  promised  his  son  2  cts.  for  the  first  example 
he  solved,  4  cts.  for  the  second,  8  cts.  for  the  third,  etc. : 
what  would  the  son  receive  for  the  tenth  example  ? 

3.  What  is  the  amount  of  $1500  for  5  years,  at  6%  com- 
pound interest  ?     Of  I2000  for  6  years,  at  7^  ? 

565.  What  is  a  geometrical  progression  ?  567.  How  find  the  last  term.  wheA 
the  first  term,  the  ratio,  and  number  of  terms  are  given  ? 


354  GEOMETEICAL     PROGRESSION". 

568.  To  find  the  Sum  of  all  the  Terms,  the  Extremes 

and  Ratio  being  given. 

4.  Eequired  the  sum  of  the  series  whose  first  and  last 
terms  are  2  and  162,  and  the  ratio  3. 

Analysis. — Since  eacli  succeeding  term  is  found  by  multiplying 
tlie  preceding  term  by  the  ratio,  tlie  series  is  2,  6,  18,  54,  162. 

(i.)  The  sum  of  the  series,    =2-1-6+18  +  54+ 162. 

(2.)  Multiplying  by  3,  =       6 +  18 +  54 +  162 +  486. 

(3.)  Subt.  (i.)  from  (2.),  486—2=484,  or  twice  the  sum. 

Therefore  484-5-2=242,  the  sum  required.  But  486,  the  last  term 
of  the  second  series,  is  the  product  of  162  (the  last  term  of  the  given 
series)  into  the  ratio  3 ;  the  difference  between  this  product  and  the 
first  term  is  486—2  or  484,  and  the  divisor  2  is  the  ratio  3  —  1. 
Hence,  the 

EuLE. — Multiply  the  last  term  hj  the  ratio,  and  divide 
the  difference  'between  this  product  and  the  first  term  hy  tlie 
ratio  less  i. 

5.  The  first  term  is  4,  the  ratio  3,  and  the  last  term  972  : 
what  is  the  sum  of  the  terms  ? 

6.  What  sum  can  be  paid  by  1 2  instalments ;  the  first 
being  $1,  the  second  $2,  etc.,  in  a  geometrical  series  ? 

569.  To  find  the  Siifti  of  a  Descending  Infinite  Series,  the 

First  Term  and  Ratio  being  given. 

Bemakk. — In  a  descending  infinite  series  the  last  term  being 
infinitely  small,  is  regarded  as  o.     Hence,  the 

EuLE. — Divide  the  first  term  hy  the  difference  betwee7i 
the  ratio  and  i,  and  the  quotient  will  be  the  sum  required. 

7.  What  is  the  sum  of  the  series  |,  J,  J,  A,  continued 
to  infinity,  the  ratio  oeing  ^?  Ans.  i}. 

Note.— The  preceding  problems  in  Arithmetical  and  Geometrical 
Progressions  embrace  tlieir  ordinary  applications.  The  others  involve 
principles  with  which  the  pupil  is  not  supposed  to  be  acquainted. 

56R.  How  And  the  sum  of  the  terms,  when  the  extremee  and  ratio  are  given  ? 
%<y^.  How  find  the  sum  of  an  infinite  descending  Beries,  when  the  first  term  and 
ratio  are  given  ? 


MENSURATION. 

570.  3£ensuratio7i  is  the  measurement  of  magni- 
tude. 

571.  3fagnitiide  is  that  which  has  one  or  more  of 
the  three  dimensions,  length,  IreadUi,  or  thicTcness;  as, 
lines,  surfaces,  and  solids. 

A  Line  is  that  which  has  lengtJi  without  breadth. 
A  Surface  is   that  which  has   length  and  breadth, 
without  thickness. 

A  Solid  is  that  which  has  le^igth,  breadth,  and  thickness. 

572.  A  Quadrilateral  Figure  is  one  which  has 
four  sides  and/(9^^r  angles. 

Notes. — i.  If  all  its  sides  are  straight  lines  it  is  rectilinear. 

2.  If  all  its  angles  are  right  angles  it  is  rectangular. 

3.  Figures  which  have  more  than/owr  sides  are  called  Polygons. 

573.  Quadrilateral  figures  are  commonly  divided  into  the  rect- 
angle, square,  parallelogram,  rhombus,  rhomboid,  trapezium,  and 
trapezoid. 

pW  For  the  definition  of  rectangular  figures,  the  square,  etc.,  scq 
Arts.  240,  241. 

574.  A    liJionibus    is    a    quadrilateral 
which   has  all  its  sides  equal,  and  its  angles 

oblique. 

575.  A  Rhomhoid  is  a  quadrilateral 
in  which  the  opposite  sides  only  are  equal,  and 
ail  its  angles  are  oblique. 

576.  The  altitude  of  a  quadrilateral  figure 
having  two  parallel  sides  is  the  perpendicular 
distance  between  these  sides ;  as.  A,  L. 

577.  A  Trapezium  is  a  quadrilateral  which  has  only  two  of 
its  sides  parallel.*     (See  next  Fig.) 

570.  What  is  Mensuration  ?    571.  Magnitude  ?  Line  ?  Surface  ?   Solid  ?  574.  A 
*hombus  ?    575.  Rhomboid  ?    576.  The  altitude  ?    577.  A  trapezium  ? 
*  Legendre,  Dr.  Brewster,  Young,  Do  Morgan,  etc. 


356  MENSURATION^. 

578.  A  Diagonal  is  a  straight  line  joining  the 
vertices  of  two  angles,  not  adjacent  to  each  other; 
as,  A  D. 

579.  The  common  measuring  unit  of  surfaces  is 
a  square,  whose  side  is  a  linear  unit  of  the  same 
name.     (Thomson's  Geometry,  IV.     Sch.) 

E^*  To  find  the  area  of  rectangular  surfaces,  see  Art.  280. 
The  rules  or  formulas  of  mensuration  are  derived  from  Geometry 
to  which  their  demonstration  properly  belongs. 

580.  To  find  the  Area  of  an  Oblique  Parallelogram,  Rhombus, 
or  Rhomboid,  the  Length  and  Altitude  being  given. 

MuUijjly  the  length  hy  the  altitude. 

Note, — If  the  area  and  altitude,  or  one  side  are  given,  the  other 
factor  is  found  by  dividing  the  area  by  the  given  factor.     (Art.  244.^ 

1.  What  is  the  area  of  a  rhombus,  its  length  being 
60  rods,  and  its  altitude  53  rods  ?  Ans.  3180  sq.  r. 

2.  A  rhomboidal  garden  is  75  yds.  long,  the  perpendicular 
distance  between  its  sides  48  yds.:  what  is  its  area? 

3.  The  area  of  a  square  field  whose  side  is  120  rods  ? 

4.  If  one  side  of  a  rectangular  grove  containing  80  acres 
is  160  rods,  what  is  the  length  of  the  other  side  ? 

581.  To  find  the  Area  of  a  Trapezium,  the  Altitude  and 

Parallel  Sides  being  given. 

Multiply  half  the  sum  of  the  parallel  sides  ly  the  altitude. 

5.  The  altitude  of  a  trapezium  is  11  ft.,  and  its  parallel 
sides  are  16  and  27  ft.:  what  is  its  area?    Ans.  236.5  sq.ft. 

582.  To   find   the   Area   of  a    Triangle,   the   Base    and 

Altitude  being  given. 

Multi2:)ly  the  hase  hy  half  the  altitude.     (Art.  539.) 

Note. — Dividing  the  area  of  a  triangle  by  the  altitude  gives  the 
lase.     Dividing  the  area  by  half  the  base  gives  the  altitude. 

578.  A  diagonal?  579.  The  common  measuring  unit  of  eurfaces?  580.  How 
find  the  area  of  an  oblique  parallelogram  or  rhombus?  581.  How  find  the  area 
of  a  trapezium?    582.  Of  a  triangle  ? 


MENSURATIOK.  357 

6.  What  is  the  area  of  a  triangle  whose  base  is  37  ft., 
and  its  altitude  19  ft.  ?  Ans.  351.5  sq.  ft. 

7.  Sold  a  triangular  garden  whose  base  is  50  yds.,  and 
altitude  40  yds.,  at  I2.75  a  sq.  rod:  what  did  it  come  to? 

583.  To  find  the  Circufnference  of  a  Circle,  the  Diameter 

being  given. 

Multiply  the  diameter  Z*?/  3. 141 59. 

^^  For  definition  of  the  circle  and  its  parts,  see  Art.  257. 

8.  The  diameter  of  a  cistern  is  12  ft.:  what  is  its 
circumference  ?  Ans.  37.69908  ft. 

9.  The  diar  .eter  of  a  circular  pond  is  65  rods :  what  is 
its  circumfer  .nee  ? 

584.  To   find    the  Diameter   of  a    Circle,  the    Circum- 

ference being  given. 

Divide  the  circumfei'ence  Ijy  3.14159. 

10.  What  is  the  diameter  of  a  circle  whose  circum- 
ference is  150  ft.  ? 

1 1.  The  diameter  of  a  circle  100  rods  in  circumference  ? 

585.  To  find    the  Area  of  a   Circle,  the    Diameter   and 

Circumference  being  given. 

Multiply  half  the  circumference  hy  half  the  diameter. 

12.  Required  the  area  of  a  circle  whose  diameter  is  75  ft. 

13.  What  '.s  the  area  of  a  circle  200  r.  in  circumference  ? 

586.  Th*;  common  measuring  unit  of  solids  is  a  cube, 
whose  sides  are  squares  of  the  same  name.  The  sides  of 
a  cubic  in»'3h  are  square  inches,  etc. 

^^  For  definition  of  rectangular  solids,  and  the  method  of  find- 
ing their  contents,  see  Arts.  246,  247,  and  284. 

583.  How/  find  the  circumference  of  a  circle  when  the  diameter  is  given' 
•584.  How  ifind  the  diameter  of  a  circle,  when  the  circumference  is  given? 
585.  How  fi'ud  the  area  of  a  circle  ?    586.  What  is  the  measuring  unit  of  solidB  ? 


358 


MEKSURATIOK. 


587.  A  Pyramid  is  a  solid 
whose  base  is  a  triangle,  square,  or 
2oolygon,  and  whose  sides  terminate 
in  a  point. 

Note. — This  point  is  called  the  vertex  of 
the  pyramid,  and  the  sides  which  meet  in 
it  are  triangles. 


588.  A  Cone  is  a  solid  which 
has  a  circle  for  its  base,  and  termi- 
nates in  a  point  called  the  vertex. 

589.  A  Frustum  is  the  part 
wliich  is  left  of  a  pyramid  or  cone, 
after  the  top  is  cut  off  by  a  plane 
joarallel  to  the  base ;  as,  a,  h,  c,  d,  e. 


590.  To  find  the  Contents  of  a  l^yramid  or  Cone,  ths 

Base  and  Altitude  being  given. 

Multiply  the  area  of  the  lase  hy  |  of  the  altitude. 

NOTH. — The  contents  of  a  frustum  of  a  pyramid  or  c^ne  are  found 
hy  adding  the  areas  of  the  two  ends  to  the  square  root  of  the  product  of 
those  areas,  and  multiplying  the  sum  hy  ^  of  the  aliitade. 

14.  What  are  the  contents  of  a  pyramid  whose  base  is 
22  ft.  square,  and  its  altitude  30  ft.  A?is.  4840  cu.  ft. 

15.  Of  a  cone  45  ft.  high,  whose  base  is  18  ft.  diameter? 

16.  The  altitude  of  a  frustum  of  a  pyramid  is  32  ft., 
the  ends  are  5  ft.  and  3  ft.  square :  what  is  its.  solidity  ? 

591.  A  Cylinder  is  a  roller-shaped.  eolicT  of  uniform 
diameter,  whose  ends  are  equal  and  parallel  circles. 

592.  To  find  the  Contents  of  a  Cylinder,  the  A.rea  of  the 
Base  and  the  Length  being  given. 

Multiply  the  area  of  one  end  hy  the  length. 

17.  What  is  the  solidity  of  a  cylinder  20  ft.  l^^ng  and 


4  ft.  in  diameter  ? 


Ans.  251.327.^  cu.  ft. 


587.  What  is  a  pyramid  ?    588.  A  cone  ?    590.  How  find  the  content^s  of  each  ? 


ME  J^SUKATIOK.  359 

593.    To  find  the   Convex  Surface  of  a   Cylinder,  the 
Circumference  and  length  being  given. 

Multiply  the  circumference  hy  the  length. 

1 8.  Eequired  the  cony  ex  surface  of  a  cylindrical  log 
whose  circumference  is  i8  ft.,  and  length  42  ft.  ? 

594.  A  Sphere  or  Globe  is  a  solid  terminated  by  a 
curve  surface,  every  part  of  which  is  equally  distant  from 
a  point  within,  called  the  center. 

595.  To  find  the  Surface  of  a  Sphere,  the  Circumference 

and  Diameter  being  given. 

Multiply  the  circumference  hy  the  diameter. 

19.  Eequired  the  surface  of  a  15  inch  globe.^Tis.  4.91  sq.ft. 

20.  Eequired  the  surface  of  the  moon,  its  diameter 
being  2162  miles. 

596.  To  find  the  Solidity  of  a  Sphere,  the  Surface  and 

Diameter  being  given. 

Multiply  the  surface  ly  \  of  the  diameter. 

21.  What  is  the  solidity  of  a  10  inch  globe?  ^.5  23.6  cu.  in. 

22.  What  is  the  solidity  of  the  earth,  its  surface  being 
197663000  sq.  miles,  and  its  mean  diameter  7912  miles? 

597.  To  find  the  Contents  of  a  Cask,  its  length  and  head 

diameter  being  given. 

Multiply  the  square  of  the  mean  diameter  ly  the  length, 
and  this  product  hy  .0034.     The  result  is  wine  gallons. 

Notes, — i.  The  dimensions  must  be  expressed  in  inches. 

2.  If  the  staves  are  much  curved,  for  the  mean  diameter  add  to  the 
head  diameter  .7  of  the  difference  of  the  head  and  bung  diameters ; 
if  little  curved,  add  .5  of  this  difference ;  if  a  medium  curve,  add  .65. 

23.  Eequired  the  contents  of  a  cask  but  little  curved, 
whose  length  is  48  in.,  its  bung  diameter  7,6  in.,  and  its 
head  diameter  34  inches.  Ans.  199.92  gal. 

591.  What  is  a  cylinder?  592.  How  find  its  contents?  594.  What  is  a  sphere 
or  globe?    595.  How  find  its  surface  ?  596.  Its  contents  ?    597.  Contents  of  a  cahk? 


3G0  MISCELLAi^-EOUS     EXAMPLES. 


MISCELLANEOUS     EXAMPLES. 

1.  A  square  piano  costs  I650,  which  is  3-fifths  the  price 
of  a  grand  piano :  what  is  the  price  of  the  latter  ? 

2.  A  man  sold  his  watch  for  $75,  which  was  5-eighths 
of  its  cost :  what  was  lost  by  the  transaction  ? 

3.  Two  candidates  received  2126  votes,  and  the  victor 
had  742  majority:  how  many  votes  had  each? 

4.  A  man  owning  3  lots  of  154,242  and  374  ft.  front 
respectively,  erected  houses  of  equal  width,  and  of  the 
greatest  possible  number  of  feet:  what  was  the  width  ? 

5.  Three  ships  start  from  New  York  at  the  same  time 
to  go  to  the  West  Indies ;  one  can  make  a  trip  in  10  days, 
another  in  12  days,  and  the  other  in  16  days:  how  soon 
will  they  all  meet  in  New  York  ? 

6.  Two  men  start  from  the  same  point  and  travel  in 
opposite  directions — one  goes  ssi  J^*  ^^  7  h.;  the  other 
27^  m.  in  5  h. :  how  far  apart  will  they  be  in  14  h.  ? 

7.  Divide  1 2 00  among  A,  B  and  0,  giving  B  twice  as 
much  as  A,  and  0  3J  times  as  much  as  B. 

8.  How  many  bushels  of  oats  are  required  to  sow 
35f  acres,  allowing  11 J  bushels  to  5  acres? 

9.  Bought  4|  bbls.  of  apples  at  $3f,  and  paid  in  wood 
at  $3|:  a  cord :  bow  many  cords  did  it  take  ? 

10.  A  mason  worked  11 J  days,  and  spending  f  of  his 
earnings,  had  $20  left:  what  were  his  daily  wages  ? 

11.  A  fruit  dealer  bought  5250  oranges  at  $31.25  per  M., 
and  retailed  them  at  4  cents  each :  what  did  he  make  or  lose  ? 

12.  Cincinnati  is  7°  50'  4"  west  of  Baltimore:  when 
noon  at  the  former  place,  what  time  is  it  at  the  latter  ? 

13.  A  grocer  bought  1000  doz.  eggs  at  12  cts.,  and  sold 
them  at  the  rate  of  20  for  25  cts. :  what  was  his  profit  ? 

14.  Bangor,  Me.,  is  21°  13'  east  of  New  Orleans:  when 
9  A.  M.  at  Bangor,  what  is  the  hour  at  New  Orleans  ? 

15.  What  is  the  cost  of  a  stock  of  12  boards  15  ft.  long 
and  10  in.  wide,  at  16  cts.  a  foot  ? 


MISCELLANEOUS     EXAMPLES.  361 

V^  1 6.  A  farmer  being  asked  how  many  cows  he  had, 
replied  that  he  and  his  neighbor  had  27  ;  and  that  |  of  his 
number  equaled  -^2  of  his  neighbors :  how  many  had  each  ? 

17.  How  many  sheets  of  tin  14  by  20  in.  are  required  to 
cover  a  roof,  each  side  of  which  is  25  ft.  long  and  2 1  ft.  wide? 

18.  A  and  B  counting  their  money,  found  they  had 
lioo;  and  that  J  of  A's  plus  %6  equaled  f  of  B's:  how 
much  had  each  ? 

19.  How  many  pickets  4  in.  wide,  placed  3  in.  apart,  are 
required  to  fence  a  garden  21  rods  long  and  14  rods  wide  ? 

20.  Bought  a  quantity  of  tea  for  $768,  and  sold  it  for 
$883.20 :  what  per  cent  was  the  profit  ? 

21.  What  must  be  the  length  of  a  farm  which  is  80  rods 
wide  to  contain  75  acres  ? 

22.  What  must  be  the  height  of  a  pile  of  wood  2>^  ft.  long 
and  12  feet  wide  to  contain  27  cords  ? 

23.  Sold  60  bales  of  cotton,  averaging  425  lb.,  at  22 J  cts., 
on  9  m.,  at  7^  int. :  what  shall  I  receive  for  the  cotton  ? 

24.  When  it  is  noon  at  San  Francisco,  it  is  3  h.  i  m. 
39  S3C.  past  noon  at  Washington :  what  is  the  difference 
in  the  longitude  ? 

25.  A  man  bought  a  city  lot  104  by  31 J  ft.,  at  the  rate 
of  $2 2 J  for  9  sq.  ft. :  what  did  the  lot  cost  him  ? 

26.  If  ^  of  a  ton  of  hay  cost  £3^^,  what  will  JJ  ton  cost  ? 

27.  What  sum  must  be  insured  on  a  vessel  worth  1 165 00, 
to  recover  its  value  if  wrecked,  and  the  premium  at  2%  ? 

28.  How  many  persons  can  stand  in  a  park  20  rods  long 
and  8  rods  wide ;  allowing  each  to  occupy  3  sq.  ft.  ? 

29.  A  builder  erected  4  houses,  at  the  cost  of  $4 2 84 J 
each,  and  sold  them  so  as  to  make  1 6%  by  the  operation : 
what  did  he  get  for  all  ? 

30.  A  man  planted  a  vineyard  containing  16  acres,  the 
vines  being  8  ft.  apart :  what  did  it  cost  him,  allowing  he 
paid  6  J  cts.  for  each  vine  ? 

31.  If  a  perpendicular  pole  10  ft.  high  casts  a  shadow 
of  7  ft.,  what  is  the  height  of  a  tree  whose  shadow  is  54  ft.  ? 

IG 


3&2  MISCELLAISTEOUS     EXAMPLES. 

32.  A  miller  sold  a  cargc  of  flour  at  20^  profit,  by  which 
he  made  $2500:  what  did  he  pay  for  the  flour? 

^S.  Paid  $1.25  each  for  geographies:  at  what  must  I 
mark  them  to  abate  6^,  and  yet  make  20^  ? 

34.  Sold  goods  amounting  to  1 1500,  i  on  4  m.,  the 
other  on  8  m. ;  and  got  the  note  discounted  at  7^:  what 
were  the  net  proceeds  ? 

35.  A  line  drawn  from  the  top  of  a  pole  36  ft.  high  to 
the  opposite  side  of  a  river,  is  60  ft.  long:  what  is  the 
width  of  the  river  ? 

36.  A  school-room  is  48  ft.  long,  ^6  ft.  wide,  and  11  ft. 
high :  what  is  the  length  of  a  line  drawn  from  one  corner 
of  the  floor  to  the  opposite  diagonal  corner  of  the  ceiling  ? 

37.  The  debt  of  a  certain  city  is  8212624.70:  allowing 
6^  for  collection,  what  amount  must  be  raised  to  cover 
the  debt  and  commission  ? 

38.  A  publisher  sells  a  book  for  62}  cents,  and  makes 
20  fo :  what  per  cent  would  he  make  if  he  sold  it  at  75  cents  ? 

39.  What  will  a  bill  of  exchange  for  £534,  los.  cost  in 
dollars  and  cents,  at  14.87!  to  the  £  sterling  ? 

40.  Three  men  took  a  prize  worth  $27000,  and  divided 
it  in  the  ratio  of  2,  3,  and  5  :  what  was  the  share  of  each  ? 

41.  An  agent  charges  s%  for  selling  goods,  and  receives 
$^35-5o  commission :  what  are  the  net  proceeds? 

42.  A,  B,  and  C  agreed  to  harvest  a  field  of  corn  for 
$230;  A  furnished  5  men  4  days,  B  6  men  5  days,  and 
0  7  men  6  days :  what  did  each  contractor  receive  ? 

43.  A  and  B  have  the  same  salary ;  A  saves  J  of  his, 
but  B  spending  $40  a  year  more  than  A,  in  5  years  was 
$50  in  debt:  what  was  their  income,  and  what  did  each 
spend  a  year  ? 

44.  If  a  pipe  6  inches  in  diameter  drain  a  reservoir  in 
80  hrs.,  in  what  time  would  one  2  ft.  in  diameter  drain  it  ? 

45.  A  young  man  starting  in  life  without  money,  saved 
$1  the  first  year,  $3  the  second,  $9  the  third,  and  so  on, 
for  12  years:  how  much  was  he  then  worth  ? 


ANSWERS 


ADDITION. 


Ex, 


Ans. 


rage  25. 

2.  T3839 

3.  18250 

4.  20000 

5.  20438 

6.  212269 

Page  26, 

1.  2806 

2.  $1941 

3.  25285  lbs. 

4.  14756  yds. 

5-  98937  r. 

6.  2051834  ft. 

7.  2460  A. 

8.  23459 

9.  185462 
10.  76876 

II-  33367 

12.  T79589 

13.  273070 


Ex. 


Asa. 


14.  2616263 

15-  9539381 

16.  $4668 

rage  27. 

17.  1376  yds. 

18.  $6332 

19.  1695  lbs. 

20.  2668  g. 

21.  10438 

22.  8636 

23.  10672 

24.  3874 

25.  15246 

26.  100980 

27.  1207053 

28.  $9193 

29.  3998  bn. 

30.  $107601 

31.  $38058 

32.  2844 


Ex. 


Ans. 


Ex. 


SS.   13800 


rage  28. 

34.  159755 
848756 
182404 
1039708 
1 1 485 
9929 
13720 

1328464 
8237027 
25148 

IIIIIIIO 

22226420 
$1460 

1925  y- 
$8190 

6987  lbs. 
93  yrs. 

$22338 


35- 
?>^- 
37- 

39- 
40. 
41. 
42. 

43- 

44. 

45- 
46. 

47. 
48. 
49. 
50- 
51- 
52. 


Page  29. 

53.  $i6829,D's; 
$33658,  alL 
156  str. 
7213  bu. 
366  d. 
$5296 

50529 
3674  A. 

$437-44 
^571-54 
$376.02 

63.  $476.19 

64.  $501.31 

65.  $475-89 
Page  SO. 

66.  $1704.28 

67.  $16988.71 

68.  $16580.34 

69.  $179403.71 

70.  $157011.73 


54- 
55- 
56. 
57- 
58. 

59- 
60. 
61. 
62. 


SUBTRACTION 


Page  35. 

2.  346 

3-  147 

4.  3106 

5.  2603 

6.  509 

Page  36. 

I-  53637 

2.  305  r. 

3.  67  lbs. 


4.  3779  y- 

5.  1719A. 

6.  11574 
7-  22359 

8.  27179 

9.  267642 

10.  235009 

11.  5009009 

12.  5542809 

13.  2738729 

14.  51989  lbs. 


15- 
16. 

17. 

18. 
19. 
20. 
21. 

22. 

23- 
24. 

25- 


309617  T. 

209354  A. 

34943 

1235993 

3633805 

33230076 

349629696 

$18990 

$1915 

$415026 

$200005 


Page  37. 

26.  $279979 

27.  iiiiiii 

1111111112 

289753017- 
746. 

270305844- 
28516. 

226637999^ 
876130. 


364 


ANSWERS. 


Ex. 


Ans. 


33'  1990005 
34.  995500 

35-  ^4564 


Fage  38. 

1.  $1200 

2.  $1107 

3.  2365  sold, 
1 1 95  left. 

4.  ^4331 
5-  4471 
6.  1279 


Ex. 


Ans. 


2,6.  999001000 

39.  2235  A. 

40.  26530000 


Ex. 


Ans. 


41.  1732  yrs. 

42.  84  yrs. 

43.  1642  y. 


Ey. 


Aks. 


44.  413000000 

45.  I3II5027 

46,  8253204 


QUESTIONS     FOR     REVIEW. 


7-  3771 

S.  803559 

9.  $14292 

10.  $963 

11.  $12523 

12.  1700 

13-  4434 

14.  112122 

15.  127680 


rage  39. 

16.  108888 

17.  126950 

18.  38022 

19.  43898 

20.  33885 

21.  762 

22.  1680 


MULTIPLICATION. 


20. 
21. 


r((ge  46. 

10224 

19705 
64896 
761824 

rrtge  48. 

15-  3563875009 

16.  2923420500 

17.  1572150300 

18.  450029050401 

19.  1924105179680 
80625  lbs. 

^55350 

22.  22360  bu. 

23.  $222984 

24.  78475  m. 

25.  1351680  ft. 

26.  $48645 

27.  $33626 

28.  $2367500 

29-  54075  yds. 

30.  $1786005 

31.  $242284 

32.  432  m. 


Page  47. 

2.  242735 

3.  1230710 

4.  3627525 

5.  20136672 

rage  50. 

3.  65100 

4.  290496 

5-  509733 

6.  3263112 

7.  145152011.111. 

8.  $295008 

9.  $197500 

Page  51. 

21.  1491000 

22.  3328000 

23.  166092000 

24.  740000  lbs. 

25.  $184000 

26.  2600000  cts. 

27.  $1050000 

28.  604800000  t. 

29.  $7350000 


Page  48. 

6.  332671482 

7.  941756556 

8.  1524183620 

9.  2751320848 


23.  2198 

24.  $4730 

25.  $522. 
?6.  $1802 

27.  $13970  B's 

$27440  C's 

28.  5673 

29.  $737 

I  30.  $18775 


10.  131247355 

11.  351410400 

12.  5321254S0 

13.  1651148750 

14.  1583318550 

30.  4022635 1915 148000 

31.  64003360044100000 

32.  22812553589100000 

Page  52. 

2,2,-   12615335010000000 

34.  312159700930000000 

35.  263200000756000000 
2^.   20776000000000 

37.  42918958404000 

38.  370475105000000 
39-  652303302651 
40.  400800840440000 

42.  17784 

43.  29484 

44.  45975 

45.  107872 

46.  454842 
47-  1585873 


ANSWERS. 


365 


SHORT    DIVISION 


Ex.    Ans. 

Ex.   Ans. 

Ex.     Ans. 

Pafje  59, 

13.  8243  h. 

2  2,  I280604-I- 

2.  218392 

14.  211  s. 

23. IOOI162 

3.  186782 

15.  8978I  r. 

24.  746367A 

4.  I72258I 

16.  $10671 

25. I2OO381 

5.  149647! 

17.  122  yds. 

26.  2346842 

6.  662107! 

18.  14140^  lbs 

27.  3562695I- 

7.  6865861- 

19.  $8121 

28.  5848142! 

8.  923808 

29.  8447232^ 

9.  922969! 
10.  5762314 

Page  60. 

30.  59363694A 

31.  64519169-A 

II.  60663768^2^ 

20.  1067102-J 

32.  37941  w. 

12.  680021033^^2 

21.  933539 

?>?>'  $6412 

Ex. 


Ans. 


34.  9065I  t. 

35.  52481-bar. 

36.  3438t'2  y^- 

37.  64692-  yds. 

38.  I9405 

39.  5812  sq.yd. 

40.  2500  hrs. 

41.  70440  b. 

42.  I14531 

43-  Z2>S'^^2>  A- 

44.  i392of  bar. 

45.  8090  cows 


LONG     DIVISION. 


Page  6.1, 

3-  2312A 
4.  4091 
5-  20761^ 


6. 

7. 
8. 

9- 
10. 
II. 
12. 

13- 

14. 

15- 
16. 

17- 

18. 


2 1 06 14 
10778/^ 
io774jf 
9759ff 

1025  2 /y 

5087 2|i 


6i294fi 

ioi77iff 

io8o88f| 

8871211 

9497off 

243f|- 

454  am. 


Page  64. 

20.  297963^ 


I99i6i|-f- 
236851-ff 


21. 

22. 

23- 

24.  450  sh. 

25.  4iiA\s.  ft. 
26. 


I2  294f|j 


27. 
28. 
29. 

30- 
31- 
2^' 
33- 
34. 
35- 

37- 
3^' 


).  $94 
54of|t  0. 
$6o2if| 
i44Hitcu.ft. 
SOyft fiy  lbs. 
8787ifll 
iii7iffff 

I2030y2^T^ 
TTr.<^T    6276 

3o878Mflfi 
I7767lfff^f 

I948II3  4 
4I332  0  3-9~5-3T2-S" 

l*a</e  65, 

$4000 


39-  $37907ffl 

40.  288mA. 

41.  $8o5oHM 

42.  $275 

43.  200 


72000 
y2y5^0TJ 


Page  66. 

2.  17. 


3. 

-is',   19  or. 

4. 

23  pounds. 

5. 

1 2  com. 

6. 

17 

7. 

23 

8. 

26 

Pcfgre  67. 

9- 

Given 

10.  961^^ 


II. 


7974H 


12.    9865^1 

13-    35776II 

14.  660421I-I 

15.  1502085^1, 

Page  69. 

24.  2283^ 

25.  4o6fSJ 

26.  iJlUU 

27.  ii3#Mi 

20.   o4:gToroo' 

00      Rft   6367 

30.  490t%^/o%^^ 

^T      620-H-i2jLI 

31.  U2O^-^0QQ^ 

3^'  h73 

^$.  229  horses 

34.  $75t¥A 

^r      £-1200] 

35-  S^-goo-^' 
36.  60  bales 


366 


AKSWEES. 


QUESTIONS     FOR     REVIEW. 


Ex.           A5fs. 

Ex.          Ans. 

Ex.           Aks. 

Ex.          Ans. 

Page  69. 

1.  146  J's; 
365  both 

2.  407  sheep 

3.  76  years 

4-  1779 

5-  3093 
6.  83li 

7.  43  rods 

8.  5 118  bu. 

9.  8613 

10.  35 

11.  200849 

Page  70. 

12.  $631  gain 

13.  302'Adays 

14.  972tV  lbs. 

15.  122-^  bar. 

16.  $1537^ 

17.  60  days 

18.  $156 

19.  io5f  yds. 

20.  $2 

21.  S9354 

22.  I5383J 

23.  252  sheep 

24.  41 J  bar. 

25-  11/^ 

26.  23833^ 

27.  482  books 

25.  13  cows 

PROBLEMS    AND    FORMULAS 


Page  73-6. 

2.  115  7  votes 
4.  II 47  votes 
6.  160  rods 


8.  31  miles 
10.  $22680 
12.  60 

14.  $2196,  ist. 
$3172,  2d. 


15.  2428,  ist. 
3136  2d. 

16.  $104  ch. 
$146  w. 


17.  45?  A's. 
30,  B's. 

18.  8269,  A's. 
$231,  B's. 


ANALYSIS, 


Page  78. 

1.  15  cows 

2.  50  lbs. 

3.  432  lbs. 

4.  60  bu. 

5.  31  lbs. 


6.  378  bu. 

7.  1 1 80  cloth; 

$10 

8.  $18 

9.  ^315 

10.  18  pears 


Page  79, 

12.  31 

13-  504 
14.  II  chil. 
16.  12  less, 
60  greater 


17- 
18. 


19. 


20. 


Given 
$112  B's. 

I361  A'& 
67,  ist. 

176,  2d. 
2II8 


FACTORING. 


Page  8S. 

23.  2  and  3 

24.  2  and  3 


25.  2  and  2 

26.  None. 

27.  2,  2,  and  2 

28.  2,  3,  and  2 


29.  5  and  3 

30.  2  and  2 

31.  2  and  2 


32.  5 

33.  2  and  z 

34.  2  and  3 


35-  I?  2,  3,  5,  7,  II,  13,  17,  19,  23,  29,  31,  37,  41,  43.  47?  53, 
59,  61,  67,  71,  73,  79,  Ss,  89,  97. 

36.  From  100-200  are  loi,  103,  107,  109,  113,  127,  131,  137^ 
139?  149?  151?  157?  163,  167,  173,  179,  181,  191,  193,  197,  199. 


ANSWERS. 


367 


CANCELLATION. 


Ex.       Ans. 

Ex.       Ans. 

Ex.       Aks. 

Ex.        Ans. 

Ex.       Ans. 

rage  90, 

8.  28 

13.    90 

18.  s^i 

23.  180  ch. 

4.   If 

9.   18 

14.  7H 

19-  378 

24.  315  bll. 

5-  8 

10.   15 

15. 84 

20.  30  bar. 

25.  28  yrs. 

6.  3 

II.     I2j- 

16.  84 

21.  51-1 1. 

26.  8Jt. 

7.  9 

12.     10 

17. 3I8A 

22.  46-f-bgs 

Page 

91. 

3. 

3 

4. 

4;  3 

,6 

5- 

6 

6. 

7 

7- 

10 

COMMON     DIVISORS. 
II.  6 


8.  5  and  3 

9.  2  and  4 

rage  93. 

2.  12 

3.  24 


rage  94, 

6.  21 

7.  15 

8.  12 

9-  3 

10.  25 


12.  4 

13.  12 

14.  2 

15.  12 

16.  2 

17.  192 


18.  one 

19-  37 

20.  2 

21.  2040 

22.  18  yd. 

23.  21  each. 

24.  8  A. 


MULTIPLES. 


Page  98, 

3.  48 

4.  84 

5.  720 


6.  600 

7.  480 

8.  330 

9.  240 


10.  288 

11.  12852 

12.  15120 
13-  73440 


14.  55440 

15.  57600 

16.  J  000 


17.  60  cts. 

18.  720  ro's 

19.  60  lbs. 


FRACTIONS. 


rage  105, 

3 
4 

5- 
6. 

7 
8 


i 

i 


121 

"2TF 

3 

2 


15-  H 


T3 


16. 


17-    "ST 


18. 
19. 

20. 

21, 
22. 

23- 

24. 

25' 
26. 
27. 
28. 


71 
"123^ 

T^ 
I 

191 

2^0" 

J 

3' 

2 

T 

3 

¥ 

2 

T 

5 

58 
TT7 


29. 

\ 

rai 

je  105, 

2. 

37 

3. 

i6i 

4. 

19 

5- 

i5i 

6. 

12 

7- 

16 

8. 

I2f 

9- 

36« 

10. 

45A 

II. 

6i 

12. 

3ll 

5^2" 


13- 
14. 

15.  i8fB 

16.  90tW 
17-    22t^J^ 
18.    2Ioi 


^o7t¥t 


46t-|5- 


19. 

20. 
21. 

22. 

24. 

25- 
26. 


^^ 


3^3^"- 


20| 

2449ff 

T  n  3  8  2  3 
^  °"4T2  0 
on-^  2  I.S 

T,3I86  T 


27.  $29f4ff 

28.  3516  yr. 

rage  106, 


108 


^      1  o  » 


3- 

4- 

5. 
6. 

7- 
8. 

9- 
10. 


9170 
~63- 
13  5  0  5 

-"T"2 

2  0  5  S  I 
—r  (TO- 
SS 7  5  5 
-IT0~ 

78fia 


4i:^Vlbl  II.  i^^ 


368 


Aif  SWERS. 


Pcifje  108 


Pa<je  111 


-  15       8 

2-  2  0?  ^F 

,  35      56      60 

3-  71)7  -5  07  TO" 


154  99  84 
2  3Tj    2TTJ  ^3T 

65  78  90 
TTQi  T^IT?  TTO 

93  5         714        462 

T3^g">  T3  (vg-j  T^ug" 


4- 

5- 
6. 

M  I98fi     2640     2970      2.16  0 

/•  T^-^0^  "5  9  4'0?  5940?  T970 

Q  14080      18480       9  2  4  0       3.80  8  0 

^*  2  46^¥0?  '24Z^iUy  ^^5^^}   2T^^0 


n      157500      25  2000      2  1262  5 

9-    T83  5  0  0>    2¥33ir(5"J    2^3T07r 

TO      I39I04      172368      10  3  488 

1°'  "5^0  81)3" 2 J  3irSl)3^J  ^^O'SITJ^ 

rage  112. 

TO   330  140  231 

^2.  T^s",  totj  Toy 

T,   74  3  6   9048   176    85  8 
^3*  TT44:?  TT44>  TTi:^?  TTi:? 
T/l   JL44   I29I5   1400 
^4'  2^2  0?  "2^X0"?  2y2^U" 
T C   1463   304  154 
^5*   2^^>  2^^>    2^5" 


16. 


20. 
21. 

22. 

23- 

24. 

25' 


594  2024   4  8 

I  I  13   1428  40 

-8-4-?  -&4-7  -g^ 

324  756 
T-44F?  T^^J 

27  40 
7-2?  7U 


384 
T4^ 


ih 


66 


60 

36   60 

T3  5  30 

-y-s-j  To 

T05   98 

,3  5  0    750    476 

loooj  Tinnr^  looo 


o. 


56     63     60 
■81F>  "8^?  "81^ 
III 
^J  4?  4" 
18     3  0     2  5 
43"?  ^T?  4T 
54         28       567 
T2^?  T2^?  TT6" 
14     15      2  0  0 
'2  0?    2  0'  "T0~ 
120      23  1      1760 
^^"^J  "2-^4'  -TS'T" 
2  5       8       7  0      175 
2  0'  "215^?  TU}  To' 


,W 


II. 


13^ 

14. 


4        4         5       103 

^TT?  ^xy?  '2Uf  Tir 

90         140       105 
^T0>     TTTT^     2TTr> 
168     180 
2T0J   2T0 


12 

S"^J 


87      2400 


I_04       5         7 
2  0  >  ^Ty>  Ti7 


T  r-  465      680     44 

16.  -f^,  ■^,  -^ 

tn  23  6  0      2690     1441? 

tQ  3  2      5  3     40 

^^'  TUj  T(J)  tu 


Ai^SWERS. 


369 


ADDITION     OF     FRACTIONS. 


Ex.  Ans. 

Ex.  Ans. 

Ex.  Ans. 

Ex.  Ans. 

Ex,   Ans. 

Page  lid. 

12.  82^V 

17.  32oi 

22.  I3t'^ 

26.  5 1 1-  y. 

13.  2T¥ff 

18.  75ifl 

23.  $7i 

27.  $489! 

8,  9.  Given 

14.  If^ 

19.  554H-S 

24.  $7 If 

29.  9t¥t 

10.  8yV 

15.  i35t2 

20.  io|| 

25.  2Mlbs. 

30.  6Jf 

II.  22 J 

16.  io3fJ 

21.  5itl 

SUBTRACTION     OF     FRACTIONS. 


Paf/c  118. 


10. 
II. 
12. 

13. 
14. 


31 

m 

1 1 

TO 


15.  2tW? 

16. 3  m 

17.  3:^ 

18.  3tV(7 

19.  2^^ 

20.  32I;  lbs. 

21.  87  2^^  A. 


Page  119. 

22.  Given 

23" 


24. 
25. 
26. 


38i 


37i 
43  f 
67f 


27.  lo2TVy 

28.  J  _ 

29.  Given 

30.  i 
31. 
32. 
33.  si 


47 

57 

TT 


53 


34. 
35- 

36.  iiIt\ 

37.  4o|gal 

38.  $291 

40.  i^ 

41.  I 


174IJ 


« 


13 
2-S 


MULTIPLICATION     OF     FRACTIONS. 


Page  121. 

18. 

$15924 

13.  643lf 

II. 

? 

2. 

$4 

19. 

I17374 

14.  1256J 

12. 

A 

3- 

$1 

3.  2j 

20. 

$45374 

15.  6i2i 

13. 

A 

4. 

1 49  A 

4.  7A 

21. 

$5418 

16.  1 009 J 

14. 

AS 

5- 

3714 

5.  i89i 

17.  21721^^ 

15. 

3il 

6. 

600 

6.  3574 

7.  583* 

Page  123. 

18.  700031 

16. 
17. 

78i 
9  cts. 

7. 
8. 

3986|-f 
i2377t'5 

8.  36 

9.  277i 

3- 

4. 

24| 

26A 

Page  124=, 

1 8. 
19. 

I36 
203il 

9. 
10. 

1731711 

'o.  48Hf 

5. 

336 

3.1 

20. 

703-i 

II. 

ii4J 

Li..    2566  J 

6. 

406 

4.  i 

21. 

1 9531- 

12. 

%of 

12.  4094M 

7. 

542^ 

5.  6f 

22. 

15352H 

13. 

4334  eta 

13.  275 

8. 

$26f 

6.  fi 

24. 

i 

14. 

$181^ 

14.  $6of 

9. 

$250f 

7.4 

25. 

4 

15. 

750  tin, 

15.  $23lJ 

10. 

$6429! 

8.  f 

3000  cop. 

16.  $I2l8J 

II. 

549fl 

9.  44! 

Page  125. 

16. 

$1918^ 

17.  iB8o4f 

12. 

1407 

10.  14 

I. 

H 

17. 

44839l§ 

870 


ANSWERS, 


DIVISION     OF     FRACTIONS, 


Ex.      Ans. 

Ex.      Ans. 

Ex.       Ans. 

Ex.       Ans. 

Ex. 

Ans. 

Page  P26. 

28. 6d>ii 

Page  130. 

3-  $i| 

II. 

ih 

A  11 

29-  315x1 

30-  ii5toV 

2.  2f| 

3.ii 

.1         T      I 

4. 17H 

5.  15  times 

12. 

13- 

50/2  bu. 

245iTy- 

4-    T35- 

7-  A 

^-    T2T 

9-  TT g- 
lo.  xMIt 
II-  tWt 

12.    ff-ill 

3i.^8i« 

6-  Hi 

14. 

86o^^2iii' 

32.  $I26J 

33.  ^IH 

4-  I^V 
5.  41-1 

7-  M 
8.  lA 

15- 

8103!- 

9-  'Ul 

10.  50  J  lbs. 

11.  3o|  A. 

12.  10  lots 
13-  42T¥y  y- 

T  4      16 

Page  133. 

Page  128. 

2.   I26f 

3-  672 
4-350 

5-375 

7.   2| 

9- A 
lo.  4j 

16. 

41  ^^--L 
120 

27  '7  s. 
'120 

13.  ^Sh 

6.  1479 

12.4 

14.  J-^ 

17. 

« 

14.    Tli?-T 

7-  447f 

13-  ^zUHh 

15-  273 

18. 

A 

15.  ^V  bar. 

8.  1645! 

'5-  ifJ 

400 

19- 

4i 

16.  ItJct 

9.  21630 

1 6-  5il 

Page  132, 

20. 

80  diws 

17.  ^/. 

10.  56727TT 

17-  3A 

I.  I43I  err. 

21. 

$4ii  " 

18.  l7f 

12.  135 

1 8-  7il 

2.  1619214:0. 

22. 

23M 

13-  477 

19.  6 

$i9A-er. 

23- 

illJ 

Page  127. 

14.  266 

20.  i4ofJ  r. 

3-  50I 

24. 

iS^ 

19-  6 A 

15.  804 

21.  lojfjs.r. 

4-  32|-  A. 

25. 

50  vests 

20.  5iM 

17- 8A 

23.  f^ 

5-  M 

26. 

44f?c. 

21.  if 

18.8 

24.  ij 

6.  3iJ 

27. 

12x^2  hr. 

22.  Ill 

19.    I2lf 

25-  2t¥? 

7.  H  sold; 

28. 

i6f 

23.   loJJI 

20.   Ilf^ 

27.tV^V 

fjown; 

29. 

60  bu. 

24-   122V0- 

22.  9 

$24783 

30- 

244f  yd. 

25.  i5tM 

23-  3tt 

l^a^re  131. 

8.  86|J  A. 

31- 

2 

26.  15HM 

24.  3-i-? 

I.  5 J  mo. 

9.  AV 

32. 

/A-V 

27-    20^Jj^ 

25-  4A 

2.  8}  lbs. 

10.  li 

33- 

3t¥5 

FRACTIONAL 

RELATIONS    OF    NU 

MBERS. 

P<^ff/e  J54. 

5-  J 

8.^ 

II.  Y^bii. 

Page  135, 

3.  h  A 

6.^ 

9-  \i 

12.  fjton 

15 

.  $253 

4- A 

7-  i 

10.  f  wk. 

13-1 

\t 

.  f59i 

ANSWERS. 


371 


Ex. 


Ans. 


17.  $132 
19. 

20. 

21. 
22.  i 

ITS- 


^00" 
3 

3 


24.  $5-- 


25- 

27- 

28. 
29. 
30. 
32. 


>$2^ 


\o29 

3  2. 
3 

9_6. 
5 

7  7_ 
~4 
200 


i4 
33^ 


Ex. 

Ass. 

33- 

if 

34- 

A 

35- 

fM 

Page  136, 

37. 

i 

3^- 

I 

39- 

I 

40. 

1 

41. 

tV 

42. 

1 

44. 

A 

Ex.   Aus. 


45-  in: 

46.  f  I  lb. 

47.  2  ft. 

48.  tV  lb. 
49-  T? 

5^-  2i50 
52.  ^'^ 

3.  84 

4.  90I 


Ex. 


Ans. 


5-  2I2i 

6.  138 

7.  1017-J 

8.  729I- 

9.  2283I 

10.  2283! 

11.  270 

12.  $14720 

13-  33250 

16.  I  ° 

17.  I 
19.  62I 


I 


Ex. 

Ans. 

20. 

83J 

J*«gre  15<5?. 

22. 

288 

24. 

9ii  ycis. 

25- 

I3  each 

27. 

iijt. 

28. 

100  cts. 

8  cigars 

30- 

1 30  J 

32- 

7  times 

34. 

3i 

REDUCTION 
2Xf/e  ^4^-       rage  144:, 

8.  4.7 

9.  21.06 

10.  84.45 

11.  93.009 

12.  7.045 

13.  10.00508 

14.  46.0007 

15.  80.000364 

17.  .06  ;  .063; 
,0109 

18.  .305;. 00021; 
.000095 

19.  .004;  .0108; 
.46;  .000065; 
.0001045 

20.  69.004; 
10.0075  ; 
160.000006 


3. 

t'A 

4. 

m 

5- 

1 

6. 

863 

7- 

iV 

8. 

100  0 

9- 

1250 

10. 

^Ww 

II. 

TODFO" 

12. 

tttJ^ 

T-7         _   _9  I 

^3*  TSiiUTT 

14*    20  0  0  0 

T  e     -JO  17  1 
^5*    iSOOOO" 

16. 


I2S00O0 

TT        I  9  8>  I 
17-   T5  0  0  0  0 

t8      15  5  0  0  7  9 
I^-    6250000 


OF     DECIM 
Page  145, 

1.  Given 

2.  .25 

3-  -4 

4-  .75 

5.  .8 

6.  .625 

7.  .25 

8.  .875 

9.  .8 

10.  .95 

11.  .6 

12.  .0875 

13.  .02 

14.  .000375 

15.  .0078125 

16.  .00875 

17.  .01125 


ALS. 

Page  146, 

19.  Given 

20.  .3333-^ 

21.  .8ss3-h 

22.  .2857  4- 

23.  .4444  + 

24.  .2727  4- 

25.  .64284- 

26.  .6041664- 

27.  .5466  + 

28.  75.6 

29.  136.875 

30.  261.68 

31.  346.8133  4-. 

32.  465.0025 

33-  523-o'^39o625 

34.  740.01375 

35.  956.0078125 


ADDITION     OF     DECIMALS 

Page  147.  I  5.  14.38916  9-  92.00537  | 

T,  2.  Given  6.  118.792  10.  37.417 

^3.  881.6217  I  7.  892.688  II.  2.3948 

4.  139.26168  ;  8.  2.76231  1  12.  23.25553  ; 


Page  148, 

13-  575-729105 
14.  53.1  bii. 

IS-  75-97  1^ 


372 

A  3S^  S  W  E  R  S  . 

SUBTRACTION     OF     DECIMALS. 

Ex.           Ans.           ! 

Ex.          Ans. 

Ex.          Ans.           1 

Ex.             /  NS. 

Bage  14=8. 

Page  Hi). 

12.    .8969755 

19.  4K/955 

I.  Given 

6.  7.831 

13.    .5496933 

20.  .000098 

2.  7.831 

7.  6.60249 

14.    .876543211 

21.  $443-825 

3.  6.60249 

8.  17.3675 

15.    .01235679 

22.  139.83  A. 

4.  17-3675 

9.  77.94794 

16.    .099 

23.  99.063 

5.   17.94794 

10.  78.569966 

17.    .00999 

24.  7-.333  ni. 

II.  2.896216     1 

18.    99.999 

107-. 40703 

MULTIPLICATION     OF     DECIMALS. 

JPage  150. 

15.  431.25  lbs. 

26.   .oooo<^<:>252 

3-  -1453 

16.  $222.9375 

27.   .OOI 

4.  .000151473 

17.  469.0625  bu. 

28.   I          [0000 1 

5.  .0000016872 

18.  $10639.75 

29.    .OOOOOOOOOO' 

6.  21800.6 

19.  .0126 

31.  3205.05 

7.  .012041505 

20.  $686.71875 

32.  8003.56 

8.  20.08591442 

2>Z'  2.43 

9.  318.0424 

Page  151. 

34.   5^8 

10.  721.36 

21.  .0025 

35.  5 

II.  .00004368 

22.  .000004 

36.  $50 

12.   1.50175036 

23.  .00000049 

37.  $600 

13.  .000721236 

24.  .000000603 

38.  27.625  bu. 

14.  .020007 

25.  .0000003 

39.  65.625  m. 

DIVISION     OF     DECIMALS. 

rage  153. 

8.  .13  + 

9.  .7115  + 

16.  10 

17.  .01 

24.  27  stoves 
26.  4-3753 

2.  .007 

10.  .9768 

18.    .000002 

27.  .063845 

3-  4 

II.  .00675 

19.    .0005 

28.  .0000253 

4.  600 

12.    .0000576 

20.     50 

29.  .0000005 

5-  -4154  + 

13.    -015 

21.    .00000027. 

30.  S.0005 

6.  30.153  + 

14.    625.5 

2  2.  46  coats 

31.  $.0475 

7.  2.4142  + 

15.     10 

^23.  44.409 +r. 

ADDITION     OF     U.     S.     MONEY. 

Page  158. 

3.    $780.25 
4.    $200.09 

7.  $17.28 

8.  $69.1925 

Page  169. 

10.  $54.50 

I.  Given 

5.    $224.53 

9.  $13131.72 

II.  $120.48 

2.  $48,625 

6.  $ 

761.4785 

12.  $76.82  X 

AKSWERSS. 

373 

SUBTRACTION     OF     U.     S.     MONEY. 

Ex.           Ans. 

Ex.           Ans. 

Ex.           Ans. 

Ex.           Ans, 

Page  159. 

5.  $585.25 

7.   $171.9625 

10.    $7.4275 

2.    $533-105    '^ 

8.  $411,075 

II.   $494,945 

3.    $604,625 

Page  160. 

9-  $.955 

12.    $28.75     : 

4.    $524.50 

6.  $948.33       1 

MULTIPLICATION     OF     U.     S.     MONEY. 

rage  100, 

Page  161. 

!    12.    $.00056 

18.    $45 

2.  $747-50 

7.  $100.9125 

13-  $5 

19.    $80.4375 

3.  $1569.24 

8.  $14.124375 

14.  $.001011 

20.    $1890 

4.  $290,625 

9.  $67.8375 

15.  $6.5625 

21.    $548,625 

5.  $60,165 

10.  $310.596255 

16.  $58,905 

22.    $178.75 

6.  $459-25 

II.  $4630.70025 

17.  $56.25 

23.    $14208 

DIVISION     OF     U.     S.     MONEY. 

rage  162, 

4.  $2.75 

5.  $2,921 

9.  .01 
10.  .1 

14.    $2,965  + 
15-     925405  + 

I.  Given 

6.   100 

II.  100 

16.     293.039  + 

2.  $1,964 

7.  361.455  + 

12.    loooo 

17.    $.0625 

3.  $.05 

8.  4000 

13.  600000 

18.    $.0547  + 

COUNTING-ROOM     EXERCISES.. 

rage  163. 

Pfige  165 

.    4.  $3183.07- 

r       6.    $5201.70 

I.  $13958.38 bal. 

2.  $206.83 

Page  166. 

7-  $717-77 
8.  $395-37 

2.  $159857. i6bal.i  3.  $4367.12 

5     5-  $71.15 

ANALYSIS. 

rage  167. 

13.  $ii4.o6J 

20.  6132  A. 

10.  $31^ 

4.  $789.25 

14.  $47.82  am t. 

21.  $478,125  1. 

II.  $291!: 

5.  $4312.50 

15.  $3007750 

2.  $66J 

12.  $35 

6.  $43l 

16.  946.1+ lb. 

3.  $65 

13-  $27f 

7-  $8924 

17.  78971b.  s'd; 

4.  $22 

14.  $107^ 

8.  $o.33j 

149  lb,  av. 

5.  $409j 

15.  $ii5f 

9.  $5,264 

18.  36.45  bu. 

6.    $I2i 

10.  $31.50 

7.  $150 

Page  169. 

ir.   102  bibles 

Page  168. 

8.  $2o8| 

17.  $210 

12.  148^  vests 

19 

I247            1 

9.  $5811 

18.  $547i 

374 


AN"SWERS. 


Ex.          Ans. 

Ex.          Ans. 

Ex.           Ans. 

Ex.          Ans. 

19.  $576 

20.  $168 

21.  8240 

23.  13.66  yds. 

24.  II 25  lbs. 

25.  1800  coc's 

26.  2690 pine's 

Pfige  170, 

2.  $78,177 

3.  $240,625 

4.  $789,625 

5.  $1169.83! 

6.  $1203.93! 

7.  $40.4875 

8.  $22.3I-J: 

9.  $100.2472 

10.  $4.51 

11.  $i4.i8J 

12.  $15,975 

3- 
4- 
6. 

7- 
8. 

9- 

(o. 

II. 


12. 

13- 
14. 

15. 

16. 

17. 
18. 
19. 

20. 
21. 
22. 

23- 
24. 

26. 
28. 
29. 


rage  100, 

614  far. 
39618  far. 
£S,  I  OS.  7d. 
i2S.  9d.  2  far. 
£41,  5s.  2d.  2  far. 
1 66 1  yds. 
£225 
4440  pwts. 

rage  191, 

7865  pwts. 
80  lbs.  7  oz. 
7  lbs.  I  oz.  i8pwt. 
18  grs. 
i49f  rings 
$27 

2653  oz. 
6721944  oz. 
2  t.  725  lb. 

II  t.  1 141  lb.  I2  0Z 

$700.55 

$280  profit 
24048  drams 
7lb.6oz.  I  dr.  2SC. 
1928495  ft. 
7  ra.  79  r.  1^  ft. 
443440  rods 


REDUCTIO  N 
Page  192, 


30. 
31. 
32. 
3Z' 
34. 
ZS- 
36. 
37. 
38. 

39- 
40. 
41. 
42. 

43- 

44. 

45. 
46. 

47. 
48. 
49. 
50- 
51- 


52. 
53. 
54. 


495782  111. 
126720  in. 
$1480 
$1556.10 
456  eighths 
2608  sixteenths 
144^  yds. 
i23|yds. 
22  vests 
$16.25  Pi'o^it 
160  r.  256  sq.  ft. 
5  A.  40  s.  r.  20  s.  y. 
697641 4 J  sq.  ft. 
10240000  sq.  r. 
10  A.  108  sq.  r. 
$2 2875  profit 
3983040  cu.  in. 
32000  cu.  ft. 
10  c.  ft.  985  c.in. 
64  C.  86  cu.  ft. 
5259  qts. 
24051  pts. 

Page  193, 

6641  bu. 

254  bu.  2  p.  3  q. 

$412.08 


55.  ^12.75 

56.  790  gi. 

57.  603  qts. 

58.  5713  qts. 

59.  34616  gi. 

60.  10834  gal.  I  pt. 

61.  168  bottles 

62.  $126.00 

6^.  2612530  sec. 

64.  176010  min. 

65-  31556929-7  sec. 

66.  9W.  6d.  2oh.i5m, 

67.  639  y.225d.5  h. 
6^.  19  y.  164  d.  6  h. 

20  min. 

69.  $212,625 

70.  13°  29'  21" 

71.  36  s.  7°  17' 

72.  855631'' 

73.  1296000" 

74.  163I:  dozen 
75-  1500  eggs 

76.  694-^  gross 

77.  9360  pens 

78.  67  lbs. 

79.  1800  sheets 

80.  41 6f  quires 


APPLICATIONS     OF    WEIGFITS    AND 


Page  194, 

1.  Given 

2.  7-i  A. 

3.  165  ft. 


Page  195, 

4.  72  rods 

5.  $46500  pr. 

6.  41 16  s.  ft. 


2  sq.  ft. 
1200  bu. 


9.  Given 


MEASURES. 

'  10.  192  bulbs 

11.  3630  gr.v. 
$190,575  3. 

12.  2\  A. 


ANSWERS. 


3T, 


Ex.           Aks. 

Ex.            Ans. 

Ex.          Ans. 

Ex.           Ans. 

Page  196. 

23.  147  b.  ft. 

24.  $278.10 

35.    $112 

2,6.  %2>Z1'^^ 

47-  i795ff  g. 

14.  $324 

37.  86751  br. 

Page  201. 

15.  $8.88 

Page  198. 

38.  $3600 

48.  56^Vin. 

16.  $402.27! 

28.  $14.85 

29.  $23.40 
30-  ^4f 

49-  57-857+ b. 

17.  ^22.50 

Page  200. 

50.  42 1 ^\  eft 

18.  I45.i3f 

40.  56  yds. 

51.  9.955|in.. 

41.  44f  yds._ 

52.  3750  lbs. 

Fafje  197. 

Page  199. 

42.  i2f  yds. 

43.  320  sods 

55-  ^7x1^ 
56.  9tVV  lbs. 

21.  i8|b.  ft.; 

32.  2|if  cords 

44.  48  yds. 

57.  871  rings 

$1.40  val. 

2,:^.  i6i  tons 

45.  640  tiles 

58.  $72H 

22.  12  J  b.  ft. 

34.  $133.20 

46.  13  J  rolls 

59.  $8o5|f 

DENOMINATE     FRACTIONS 


Page  202. 

2.  f  qt. 

3.  If  d. 

4.  2^  hr. 

5.  If  oz. 

6.  y^^-  s.  in. 


Page  203. 


8.  $ 

9.  £-J 


T5"0~(J 


TTJ?" 


TTT2 


oz. 


II-  tIo  gal. 


12. 


m. 


3-   16000  t- 


16.  13s.  4d. 

17.  3  pk.  6  qt. 

18.  1250  lbs. 

19.  10  oz.  10  p. 

20.  3  fur.  13  r.  I 
yd.  2  ft.  6  in. 

21.  146  s.r.  2os.y. 
I  s.  ft.  72  s.  in. 

22.  112  cu.  ft. 

24.  n  gal. 

-,-      T  229    IV, 

26.  11  bu. 

27.  2Vx)  ton 

28.  Iff  sq.  yd. 

29-  tW^ 


30- 


2  3_3.  ^J.. 


504 

c. 

32.  M 

34.  2S.  6d.o.4  + 
far. 

10  oz.  18  p. 
22.56  grs. 
2  furlongs 
32  rods. 
2  qt.  I  pt. 

38.  39'  36" 

39.  I  d.  i6h.29 
m.  16.8  s. 

40.  5051b.  1.92 
oz. 


31.  i 


35- 
37 


41.  17.28  grs. 

42.  £5,  i2s.  6d. 
0.4032  far. 


Page  206. 

44.  .5423+  lb. 
.005  ton 
.45539 +  ni. 
.04  lb. 
.5bl. 
.409+  r. 

•25 

.48  bl. 
1.44  r. 
.166  +  wk. 
.28182  + 
.41666 +  0. 


45 
46 

47. 
48. 
49. 

50- 
51- 
52. 
53- 

54. 
55. 


METRIC     NOTATION     AND 

Page  212. 

5370.9845  dekameters; 
537.09845  hektometers ; 
53.709845  kilometers; 
537098.45  decimeters. 


NUMERATION. 


2.  450.5108  dekagrams; 
450510.8  centigrams; 
45.05108  hektograms; 
4.505108  kilograms. 


376 


AN"SWERS, 


REDUCTION     OF     METRIC     WEIGHTS     AND     MEASURES. 


Ex 

AN8. 

Ex 

Ans. 

Ex. 

Ans. 

I. 

2. 

3. 

4- 

rage  213. 

Given 

437500  sq.  m. 
867000  grams 
26442  liters 

5- 
6. 

7. 
8. 

9- 

256100  sq.  m. 
865  2000  cu.  dm. 
4256250  grams 
Given. 
65.2254  hektars. 

10. 
II. 
12. 

0.087  kilos. 
1.48235  kg. 
39.2675  kl. 

APPLICATIONS    OF    METRIC    WEIGHTS    AND     MEASURES 
rage  215. 

3.  39.14631  m. 

4.  1 9.8 13 1 J  gals. 

5.  15.89  bu. 

6.  4.2324  oz. 

7.  303.68365  lbs. 


8.  Given 

9.  148.87775  A. 
10.  4237.92  cu.  ft. 

12.  58.2934-  m. 

13.  6236.959+  kg. 

14.  236.585  +  liters. 


15.  72.492+  kl. 

16.  143.223+  kg. 

17.  6000.06+  s.  m. 

18.  16.378+  hect. 

19.  410.748+  c.  m. 

20.  27958.715 +c.m 


f. 


Page  217. 

3.  £11,  los.  od.  2  f. 

4.  26T.3cwt.83lb. 
3  oz. 

45  bu.  o  pk.  2  qt. 
74J  yds. 
1093  lb.  5  oz. 
55  gal.  2  qt. 


COMPOUND     ADDITION. 


9.  196  bu.  2  pk.  7  q. 

10.  6  C.  80  cu.  ft. 

11.  98  bu.  3  pk.  2  q. 

rage  218, 

12.  Given 

13.  23  wk.  I  d.  17  h. 

58  m. 


14.  64A.  7s,r.  io|s.y 

17.  i5Cwt.5olb.  40Z 

18.  2  pk.  4  qt. 

19.  9  oz.  ipwt.  logr. 

20.  3  s.  6  d.  ,^.2  far. 

21.  2  A.  52  sq  r. 

22.  2  0.  81  en   ft* 
1209!  cu.  in 


COMPOUND    SUBTRACTION. 


Tage  219, 

2.  I  fur.  39  r.  I  yd.  2^  ft. 

3.  7  lb.4oz.  17  p.  9grs. 

4.  7  T.  8  cwt.  26  lb. 

5.  8  gal.  I  qt.  ipt.  2gi. 


Tage  220. 


6.  159  A.  12  sq.  r. 
222^  sq.  ft. 

7.  46  cu.  ft.  1689  cu.  in. 

8.  1 45  A.  2 1  sq.  r. 

9.  46J  yd. 


10.  3  m.  5  fur.  38  r. 
5  yd.  o  ft.  I  in. 

2.  9  s.  9  d. 

3.  2pk.5qt.  i.2p, 

4.  4  J  pt. 

5.  I  lb.  2  oz.  2  p. 

6.  1 14.4  sq.  r. 

7.  31.25  lbs. 

rage  221-2, 

3.  67  y.  9  m.  22d. 

2.  113  days 

3.  74  days 


4.  150  days 

5.  224  days 

6.  loi  days 

Page  223. 

10.  86°  19'  24" 

11.  7°  24'  7" 

12.  18°  2' 

13.  5°  6'  46" 

14.  15°  4'  16" 

15.  19^"  35' 

16.  56°  II' 

17.  70°  29' 

18.  10°  19' 38" 


ANSWERS. 


377 


COMPOUND     MULTIPLICATION. 


Ex. 


Ans. 


rage  224=. 

2.  98T.i7CWt.28lb. 

3.  £151,  15s.  9jd. 

4.  33 oz.  15 pw. log. 

Page  225. 

7.  331  gal.  2  qt. 


Ex. 


Ans. 


8.  22  C.  91  cu.  ft. 

9.  £23,  15s.  3|d. 

10.  562  m.4fu.  241'. 

11.  1937  bu.  I  pk. 

12.  22  0.  57  cu.  ft. 

13.  61  T.  844  lbs. 

14.  1307  r.  8  qr.  8  s. 


Ex. 


Ans. 


15.  161°  37'  30'' 

16.  431  h.  15  m. 
17-  ^ZZ  sq.  r.  21  sq. 

yd.  f  sq.  ft. 

18.  73  T.  1492  lbs. 

19.  15  7 1  bu.  2  p.  4q. 
—  1946  gal.  3  q.  I  p. 


20 


COMPOUND     DIVISION. 


rage  227. 

3.  4  fur.  8  r.  2  yd.  2^  ft. 

4.  6  gal.  3  qt.  o  pt.  3j  gi. 

5.  25  bu.  o  pk.  I  qt.  f  pt. 

6.  15  A.  106  sq.  r.  5  sq.yd.  2^ 
sq.  ft. 


7.  26f  spoons 

8.  8800  rails 

9.  1049/y  times 

10.  12  books 

11.  2  A.  64  sq.  r. 

12.  6  bu.  I  pk.  I  qt. 


COMPARISON     OF    TIME    AND     LONGITUDE. 

rage  228. 

1.  Giyen. 

2.  43  m.  32.13+  s. 

3.  1 1  o'c.  1 2  m.  4  s. 

4.  i2o'c.37m.  8.4  s. 

5.  49  m.  20  s. 


6. 

I  o'c.  30  m.  E. ; 

10. 

6°  9' 

10  o^c.  30  m.  W. 

II. 

6°  45'  5" 

7- 

54  m.  19.8  sec. 

12. 

45°  24'  45" 

8. 

13111.44.86+8. 

13- 

56  m.  49+  sec> 

rage  229. 

14. 

12  o'clk.  34  HL 

9- 

Given 

47  sec. 

PERCENTAGE. 
Fage  232. 

13-  -50;  -25;  -75;  -20;  .40;  -60;  .80 

14.  .10;  .70;  .90;  .05;  .35;  .12;  .06 

15.  .661;  .i6|;  .125;  .625;  .875;  .58J;  .91^ 


rage  234. 

3.  $24.21 

4.  10.8  bu. 

5.  22.6  yds. 

6.  8.64  oxen 

7.  8.25  yds. 

8.  $11,584 


9.  100.8  lbs. 

10.  63.14  T. 

11.  52.5  men 

12.  145.656  m. 

13.  $857,785 

14.  £.106 

15.  1.25  1. 


16.  1.38  k. 

17.  840  lbs. 

18.  $1000 

19.  78.75  bu. 

20.  .35  lbs. 

21.  9.765  gals. 

22.  840  men 


24.  $215 

25.  i57.'2  lbs. 

26.  32.25  m. 

27.  116  1. 

28.  580  sheep 

29.  156  bu. 

30.  $925 


378 


AKSWEKS. 


Ex.          Ans. 

Ex.           An3. 

Ex. 

Ans. 

Ex.          Ans. 

,51.  576  CU.  ft. 

14.    $10200 

-Pa</e  238, 

19.  3000  m. 

32.   £2,  IS. 

15-   I9750 

4- 
5- 

250  bu. 

$362.50 

20.  360000  p. 

Page  236. 

JPage  237. 

6. 

180  tons 

Page  24:0. 

3.  1 150  s.  C; 

2.   i2>Wo 

7. 

£450 

5.  809 

S80  s.  D. 

3-   ^S% 

8. 

600 

6.   1820 

4.  $2300.91 

4.  zzWo 

9- 

360  fr. 

7.  Given 

5.  $2850 

5-  ^S% 

10. 

8000 

8.  f 

6.  504  bales 

6.  gi% 

II. 

$200 

9.  T%  B's 

7.  2018.75  gal. 

7.  2>\% 

12. 

$i6666f 

10.  5175  m. 

8.  13 1 2  m. 

8.   10^ 

13- 

$17600 

II.  $3648 

9.  $47040 

9.  80^ 

14. 

54.4  yds. 

12.  6875  p. 

10.  129  t. 

10.  10^ 

15. 

$50  " 

13'  $3350 

II.  $1410.50     - 

II.  2M% 

16. 

100 

14.  228  sheep 

12.  $228 

12.   42f^ 

17- 

$5000 

15.  400  pupils 

13.  $4823.33^ 

13.  3o|i^        . 

18. 

$1920 

16.  5800  m. 

COMMISSION     AND     BROKERAGE. 


Page  242. 

Page  244. 

3.  $31-14315 

12.  $9000 

4.  $366,225  com. ; 

13.  $4212  sales; 

$10902.225  paid 

$4001.40  net  p. 

14.  $1500  col.; 

Page  243. 

$1432.50  paid 

5.  $4310.145 

15.  $9000  sale ; 

6.  $238.66! 

$8865  rec'd 

8.  i% 

17.  $5100 

9-  5% 

18.  $6834.872  sales; 

10.  i% 

$170,872  com. 

PI 

lOFIT    AND     LC 

Page  247-9. 

II.   $453.60 

3.  $22.20 

12.  $2942.50 

4.  $22 

13.  $9520 

5.  $.0875 

14.  1 1^  cts. 

6.  $16.65 

15.  $1  a  pair- 

7.  $1.40 

16.  $2.77^  eacli 

10.  $2925  A.; 

17.  $4.90  a  yd. 

$2075  B. 

18.  $4331.25 

Page  245,  6. 

19.  $15488.89  + 

20.  $26635.294  + 

22.  $3010.75 

23.  $2419.234- 

24.  $4926 1.083+ in.; 
$738,916+  com. 

25.  $13687.50  g.  a.; 
$360.75  ch.; 
^13326.75  net  p. 

26.  $12127.83  net  p. 


19. 
21. 

22. 
23. 
25. 
26. 
27. 
28. 


$12810 

28f^ 

60% 
100% 


Ao% 


ANSWERS. 


379 


Ex 

AN9. 

Ex. 

Ans. 

Ex. 

Ans. 

rage  250, 

rage  252, 

14. 

$7  cost; 

29. 

so% 

51- 

$.736  marked  p. 

$8.05  selFg  pr. 

30- 

Given 

52. 

I34.78+  m.  pr. 

15- 

$2  cost; 

31- 

So% 

53. 

$4.80  m.  pr. 

$1.75  sell'g  pr. 

32. 

50^ 

54. 

$1.1 6f  m.  pr. 

33- 

33i% 

55. 

$75  each 

Page  254, 

34. 

$12.60  pr.; 

iW/o 

56. 

^5-75  per  yd. 

16. 

$5200  sales; 
$4940  net 

36. 

$9750 

Page  253. 

17. 

$24000  sales 

I. 

$2,691  prof. 

18, 

$1.25  cost 

Page  251, 

2. 

$6356.72^ 

19. 

$5250  cost; 

39- 

$1  cost  per  lb. 

3. 

^3487-575  sh.; 

$1750  gain 

40. 

$22500  cost; 

$850.84  ging.; 

20. 

.7894^1  per  gal.; 

$30000  selFg  pr. 

• 

$4338.415  both 

.  1 48 1  gam  per  g.; 

41. 

1 1 7500  spend; 

4. 

$22  loss 

$49.73  wh.  cost; 

$2 1 000  amt.  sales 

5- 

$77812.50  cost 

$9.33  wh.  gain 

42. 

$12000  A's  in  v.; 

6. 

$897.45  net  pr. 

21. 

$10  a  barrel 

$9375  B's  inv. 

7. 

38f^ 

22. 

$51314  cost; 

43- 

$250000  cost; 

8. 

5 6 J^  prof.; 

$6413!  gain 

$275000  amt.  s. 

$3240  prof. 

23- 

$15798.963  sale; 

44. 

$.02  cost; 

9- 

i6|-;?^  loss 

$552,963  com. 

$.025  selling  pr. 

10. 

•052  or  5^-^c.; 

24. 

38A^ 

45* 

Given 

$1766.835  net  p. 

25. 

S^%  loss 

46. 

$10061.71^ 

II. 

25^  loss 

26. 

$60  m.  p. 

47- 

$13319.672 

12. 

125^ 

27. 

6\%  loss 

48. 

£22  cost 

13. 

$9300  cost ; 

28. 

15  cts. 

49- 

3  cents 

$11160  sell'gpr. 

29. 

iot?o^1oss 

INTEREST. 


Page  258. 

3-  ^2.54 

4.  $1,676 

5.  $1,224 

6.  $4,523 

7-  $43-356 
8.  $110.77 
9-  $33-42 

10.  $23,364 

11.  $25,577 

12.  $94.35 

13.  $17.91 


14.  $680.28 

15.  $81.44  + 

16.  $2875.792 

17.  $362,315 

18.  $2759.962 

19.  $5032.083 

20.  $122.40 

21.  $226.69 

22.  $45,053 

23.  $216,489 

24.  $493-2o 

25.  $3923.47 


27.  $99.96 

28.  $24.00 

29.  $163,842  ; 
$2740.652 

Page  259. 

2.  $24.44 
3-  $6,252 

Page  260. 

4.  $4.82 

5.  $6.75 


6.  $73.96 

7.  $64.18 

8.  $26.49 

9.  $1.65 

10.  $29,933 

11.  $60.04 

12.  $30.52 

13.  $2450.80 

14.  $20819.80 

Page  261, 

3-  $2.8435 


80 


ANSWERS, 


Ex.    Ans. 

Ex.     Ans. 

Ex.    Ans. 

Ex.    Ans. 

4.  $2,192 

4.  6^ 

7.  14-f-yrS. 

.  2.  $235.85 

5-  Iii-4i3 

5.  8^ 

8.  i6f  yrs. 

3-  ^327-36 

6.  $13.89 

6.  9t\% 

9.  18  J  yrs. 

4.  $8928.57 

7.  ^1285.33 

7.  8i^ 

5.  $892.86 

^8.  $13,876 

8.  5^;  10^ 

2.  $i666| 

^  6.  $5582.142 

9.  $4363.044 

9.  7^ 

3.  I3000 

10.  $ii4.i6f 

10.  6^ 

Page  268,9, 

II.  $185.18 

Page  264, 

I.  Given 

12.  $81,358 

Page  263, 

4.  I2333-J 

2.  $1519.71 

3-  5f  y->  or  5 

5.  $2500 

3.  %  38.63 

Page  262, 

y.  8  m.  1 7  d. 

6.  $40000 

5.  $270.19 

2.  {% 

4.  9  m.  2  d. 

7.  $10000 

6.  $388.23 

3-  1% 

5.  J  y.,  or  3  m. 

8.  $25000 

7.  $516.32 

COMPOUND     INTEREST 


Page  273,  4, 

1.  Given 

2.  $112.52 

3.  $161.63 

4.  $164.61 


5.  $200.63 

6.  $2165.713 

7.  Given 


9- 
10. 
II. 
12. 


%5i-58 

$1377.41 

11543-65 

$8104.25 

$32564.58 


13.  $21825.26 

14.  $29849.56 

15-  ^37704-95 
16.  $3298.77 


DISCOUNT. 


Page  277. 

1.  Given 

2.  $283.47  + 

3.  $462.96 

4.  $1213.59 

5.  $2336.45 

6.  $4464.28 


Given 
$86.57 
$101,892 
$91.35  dis; 
$2283.65  p. 

11.  '^3775-264 

12.  $20,377 


7- 
8. 

9- 
10. 


13.  $528.33 

14.  $46.73  for. 

Page  279. 

3.  $718,685 

4.  $989.50 

5.  $1721.875 


6.  $17.25 

7-  ^735-70 

8.  Given 

9.  $768.44 

10.  $1578.81 

11.  $2226.23 

12.  $7503.83 


Page  282. 

2.  $243 

3.  $525 

4-  1^525 

5.  $400 

6.  $476 


STOCKS 

8.  $12880 


AND    BONDS 


Page  283. 

9.  $10497.50 

10.  $6^6^.61 

11.  $6336.11 


12.  $10852.37^ 

13.  $19293.75 

14.  $672  cur. 

16.  15^ 

17.  H% 

18.  6i% 


Page  284. 

20.  40  bonds. 

21.  $12000 

23.  $44i66| 

24.  $35625 

25.  $56000 


A  N  S  W  E  K  S  . 


m 


EXCHANGE. 

Ex.           Ans 

Ex.           Ans. 

Ex.           Ans. 

Ex.           Ans. 

B^({/e  '4H7. 

10.    $2617.801 

rage  290, 

II.  J2l542,IIS. 

2.    $2049 

II.    $3751.95 

6.  $56125.124- 

6d. 

3.    $3488.80 

12.    $3750 

7.  Given. 

4.    I4 1 30.647 

8.iE5ii,  15  s. 

JPagre  J?<>i. 

5.    $203 

Tage  289, 

4  d.  3.2  far. 

12.  Given 

8.  $1219.51 

I,  2.  Given. 

9.  £774,  7  s. 

13.  $675.68 

3.  $1858.80! 

lod.  1.28  far. 

14.  Given 

Taqe  288, 

4.  $4866.^0 

10.  i8io26,  8  s. 

15.  13050.00  f. 

0.  $1491.053 

5.  $56.62ig. 

7.2  d. 

16.  16474.50  f. 

INSURANCE. 


rage  293, 

1.  Given 

2.  $6.65 

3.  $38.40 

4.  $84,375 


5.  $10.70 

6.  $25.00 

7.  $197.60 

8.  $875 

9.  $1569.625 


10.  Given 

11.  $15747.423 

12.  $27806.122  + 
^3-  $37105-263  + 


1.  Given 

2.  $125.00 

3.  $2437.50 

4.  $5000 

5.  $45000  gr. 


rage  297. 

$1003.75,  C'stax 
$1250,      D's  " 
$1375,      E's  " 
$925,        Fs  « 


Page  299, 

2.  $4062.50 

3.  $1063.314 


TAXES. 

2%  rate; 
$159.75,  A's  tax 
Z%  I'ate ; 

$450,  G's  tax 


6.  $309.75  H's  tax  j  II.  $11052.63  + 


7.  5^  rate; 

$170,  A's  tax 

9.  $3645-831- 
10.  $5507.853  + 


DUTIES 
4.  $1999.20 

6.  $945 

7.  $2296.80 


Page  300. 

2.  $118.25 
3-  ^425.75 


EQUATION    OF    PAYMENTS. 


Page  302. 

2.  4  m.  from  J.  20,  or  Oct.  20 

3.  8  m.  18  d. 

4.  Moll.  10+17  d.=:Mch.  27 

5.  June  I,  or  in  3  m. 

AVERAGING 
Page  306. 

1-3.  Given 
4.  7  m.  extension ; 
Bal.,  $1650 


Page  303, 

6.  Given 

8.  July  15+51  d.=Sept.  4 

9.  $1483.25  ami; 

42.9  d.  av.  time;  due  Nov.  17 

ACCOUNTS. 
5.  $300  bal.  debts ; 
22320  bal.  prod. ; 
74  d.  av.  time 
Due  May  23. 


38^ 


AFSWEES, 


SIMPLE     PROPORTION. 


Ex. 


Ans. 


JPage  S12, 

3.  $160 

4.  I165 

5.  $450 

6.  $6 

7.  £i9,7s.6id. 

9-  135  A. 

10.  $93 7 J 

11.  270  miles 

12.  $288 

13.  $822! 


Ex. 


Ans. 


14-  ^T-33 

15.  2240  times 

16.  5  years 

17.  8  hours 

rage  314. 

18.  iioomen 

19.  240000  lb. 

20.  27  horses 

21.  $612 

22.  £ij- 

23.  I5.06J 

24.  I880.40 

25.  $490.52^ 


Ex. 


Ans. 


26.  $5223A\ 

27.  $12.50 

28.  14.4  in. 

29.  26f  yds. 

30.  97920  t. 

31.  180° 

32.  80  ft. 

T,^.  64!  orang. 

34.  3000  mi. 

35.  33  men 

36.  30  ft. 

Page  315, 

37'  ^1920 


Ex. 


Ans. 


38.  25  min. 

39.  6600  rev. 

40.  $40 

41.  2 1  lirs. 

42.  3x7-1^^8. 

43.  -45  days 

44.  1000  rods 

45.  300  hrs. 

46.  $191.78 

47.  8  m.  51  s. 
before  9 

48.  3^  cts.  loss 

49.  150  m.  less; 
186  m.  gr. 


COMPOUND     PROPORTION 


Page  317. 

3.  2f  days 

4.  9  horses 

5.  33  J  weeks 


JPage  3 IS. 

6.  |i26.66f 

7.  $100 

8.  360  miles. 


9.  I750 

10.  10285!  yd. 

11.  374  days 

12.  $i454A 


13.  750  lbs. 

14.  $62.50 

15.  240  sofas 

16.  864  tiles 


PARTITIVE     PROPORTION. 


rage  319. 

40  s.:  60  s. ;  100  s. 


4.  60  bn.  oats ;  80  b.  p. ;  no  b.  c. 

5.  71;  io6i;  142;  i77iA. 


PARTNERSHIP.— Paj7e  321-3. 


2.  $2  2  2f,  A's  loss; 
$277j,  B's     " 

3.  $340,  A's  share ; 
$510,  B's      " 

4.  l5o67|f,A'sg'n; 
$4054^.  B's  " 
^ZZim,  C's  « 

5.  $3400;  I5100; 
and  I6800 

6.  $64,  A's  share ; 


rage  324. 

2.  $1687.50,  A's  sh.; 


$96B's;$i6oC's 

8.  $142.85!,  A'ssh. 
$266.66f,  B's  " 
$190.47-11,  C's" 

9.  $24.48^1,  one ; 
$25.51^^,  other 

10.  $424^^,  A'ssh.; 


$3i8Jf,  C's  " 

$282ff,  D's  " 

BANKRUPTCY. 

$i2o5.35f,B'ssh.; 
$857.i4f,D's  « 


11.  $1500,  HU; 
$2250,  Cont; 
$3000,  Am. 

12.  $263.38ifi|,A's; 
$447.76tVAB's; 
^69i.39AVV.C's; 
$io97.45A%V,D. 

13.  $402iffi,  A's; 
$3846^11,  B's; 
^4I3It¥j.  C's 


3.  $2060,  D  rcc'd  ; 
$12100  net  proc. 


ANSWERS. 


383 


ALLIGATION 


Ex. 


Ans. 


rage  325. 


2.  151 


cts. 


3-  95  A  f'-ts. 

4.  9JI  cts. 

5.  2o|f  carats 

6.  $15.68 

Page  327- 

8.  35  bu.  ist; 
40  bu.  2d ; 
40  bu.  3d 


Ex. 


Ans. 


9.  2  p.  at  15; 

1  p.  at  18; 

2  p.  at  2 1 ; 

5  p.  at  22 

10.  9  lb.  at  3  2  c; 

6  lb.  at  40 ; 
6  lb.  at  45 

11.  8  lb. at  20  c; 

3  lb.  at  27; 
5  lb.  at  35-; 
1 2  lb.  at  40 


Ex. 


Ans. 


Page  328. 

14.  3  J  lbs.  ea. 

15.  5oq.at4C.; 
300  q.  at  6 

Page  329. 

17.  20  lb.  at  6  c.; 
20  lb.  at  8 ; 
60  lb.  at  12 


Ex. 


Anb. 


18.  54yyb.  28  c; 

18^  lb.  30; 

54Alb.  38; 

72Alb.  42 
For  other  ans. 
see  Key, 
i9-53Tg-4oc.; 

53i  g.  45  ; 

53ig-  50; 
140  g.  60 


INVOLUTION  —Page  331, 

8.  25;  36;  49;  64;  81;  100;  400;  900;  1600;  2500;  3600, 

4900;  6400;  8100. 

9.  .25;  .36;  .49;  .64;  .81;  .0001;  .0004;  .0009;  .0016;  .0025; 

.0036;  .0049;  .0064;  .0081. 


10.  125 

11.  64 

12.  2299968 


13.  1024 

14.  4096 
15-  15625 


16.  8.365427 

17.  64.014401080027 

18.  64024003.000125 


20.  Iff 

21.  ^2¥^ 


EXTRACTION  OF  THE  SQUARE  ROOT. 


Page  338. 

5.427 

6.  719 

7.  772 

8.  1.892  + 

9.  .64 

10.  .347 

11.  .41 


12. 

.8514+ 

13. 

•355  + 

14. 

1.635  + 

15. 

69.47  +- 

16. 

21.275  + 

17. 

2.236  + 

18. 

2.64  + 

19. 

2.828  + 

4 


Page  339-4 1. 

2.  420  rods. 
3-  559.28  r. 

4.  119  trees 

5.  238  men 

7.  108  yds. 

8.  30  rods 


20.  3.16  + 

21.  3.316  + 

22.  3.46  + 

23.  4.42 

24.  57-3 
25-  9-36  + 

26.  mil. II  + 

27.  7856.4 


APPLICATIONS. 

9.  438.29+  m. 

10.  103.61  +  ft. 

11.  56.56+  ft. 

12.  75.81+  ft. 

Page  342. 

15.  St  min. 


28. 

98.7654 

Page  339 

32.  H 

33'  -745  + 
34.  .866  + 

35. 
3^' 

2.529  + 
3-^3  + 

16 

24 
30 


37.  4.1683 
38' 

39' 

40.  .8545  + 

41.  i.oi8  + 
42.^ 

43.  ¥ 


16. 

17. 

18. 
19. 

20. 
21. 
22. 


Given 
12 

54 
63.49  + 

1-75 

8 
^^ 

72 
TTCF 


ANSWERS. 


EXTRACTION     OF    THE    CUBE    ROOT 


Ex.         Ans. 

Ex.          Ans. 

Ex.           Ans. 

Ex.          Ans. 

l*nge  348. 

7.    2.76 

13.  129.07+ in 

Page  34=9. 

1-2.  Given 

8.  2.57  + 

14.  Given. 

I.  Given 

3.  85 

9.   8904 

16.  .746  + 

2.  52  ft. 

4.  4.38 -f 

10.  .632  + 

17.  f* 

4.  16  ft. 

5-  72 

II.  85.6+ yds. 

18.  A 

5.  462.96 +  c  f. 

6.  1.44  + 

12.  15.3+  ft. 

19-  4.3  + 

6.  3.072  tons 

ARITHMETICAL 

5.  I360  ^ 
II  children 


15 
8 


i  7. 


PROGRESSION. - 
9.  3  yrs. 


10.  Given 


II.  78  strokes 


GEOMETRICAL     PROGRESSION. 


Page  353, 

1.  96 

2.  I10.24 


3.  .^2007.3383664; 
$3001.460703698 

4.  242 


Page  S54, 

5-  1456 
6.  1^4095 


MENSURATION  .—Page  350-9, 


1.  Given 

2.  3600  s.  y. 

3.  144008.!'. 

4.  80  rods 
7.  $90.90^ 


9.  204.20335  r. 

10.  47.746-mMf  ft. 

11.  3i.83i3fl«^r. 

12.  4417.8609375  s.  ft. 

13.  3183.1  s.  r. 


15-  3817-03185  cu.  ft. 

16.    52  2f  CU.  ft. 

18.  756  sq.  ft. 

20.  14684558.20796  5.  m. 

22.  26065 1 609333 J cii.m. 


M  I S  C  E  L  L 

1.  $1083! 

2.  I? 1 20  cost; 
$45  loss 

3.  692  less; 
1434  great. 

4.  22  ft. 

5.  240  days 

6.  143^  miles 

7.  $20,  A's; 
$40,  B's ; 
^140,  C's 

8.  79-59¥bu. 

9.  5.26^%  C. 

10.  |;3 

11.  |i 64.06 J  c. 


AN  ECUS    EXAMPLES.— JPrtflre 

^45-93j  pr. 

21.  150  rods 

Z^' 

12.  12  o'cl.  31 

22.  8  ft. 

37.^ 

m.  20^  s. 

23.  I6038.72 

38. 

13.  $120  cost; 

24.  45°  24' 45" 

39- 

830  profit 

25.  ^8125 

40. 

14.  35  m.  8  s. 

26.  £3#f 

past  7  A.M. 

27.  $16836.734 

15.  I24 

28.   14520  per. 

41. 

16.  3 farmer's; 

29.  $19878.92 

42. 

24  neigh  Vs 

30.  $680,625 

17.  540  sheets 

31.  77ift. 

18.  $40.  A's; 

32.  $12500 

43- 

$60,  B's 

Z2>'  ^1-59 

19.  1980  pick. 

34.  $1446.625 

44. 

20.  15^ 

35-  48  ft. 

45- 

360-2, 

61  ft. 
I226196.489 

44^ 

$2607.291 
$5400, 1  St.; 
$8100,  2d.; 

^13500?  3^^ 
$2574.50 
$50,  A; 
^75,  B; 
$105,  0 
$120  each 
$130,  B 
5  hrs. 
$265720 


^4 


VB  35846 


M249549 

QMOX 

EDUC. 
THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


